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F R C F <br />
• F + 0<br />
• F \ {0} · 1<br />
• a(b + c) = ab + ac a, b, c ∈ F<br />
V F <br />
V +<br />
+ : V → V (v1, v2) ↦→ v1 + v2 ∈ V <br />
v1, v2, v3 ∈ V (v1 + v2) + v3 = v1 + (v2 + v3)<br />
v1, v2 ∈ V v1 + v2 = v2 + v1<br />
0 ∈ V v + 0 = v v ∈ V <br />
v ∈ V −v ∈ V −v + v = 0<br />
· V <br />
· : F × V → V (λ, v) ↦→ λv<br />
λ ∈ F, v1, v2 ∈ V λ(v1 + v2) = λv1 + λv2<br />
λ1, λ2 ∈ F, v ∈ V (λ1 + λ2)v = λ1v + λ2v<br />
λ1, λ2 ∈ F, v ∈ V (λ1λ2)v = λ1(λ2v)<br />
v ∈ V 1v = v<br />
V F v ∈ V, λ ∈ F <br />
0 · v = 0 λ · 0 = 0<br />
−v = (−1) · v<br />
λv = 0 λ = 0 v = 0<br />
0 · v = (0 + 0) · v = 0 · v + 0 · v 0 = 0 · v<br />
−v + v = 0 = 0v = (−1 + 1)v = (−1)v + 1v = (−1)v + v −v = (−1)v
λv = 0 λ = 0 λ −1 λ −1 (λv) = λ −1 0 = 0 <br />
λ −1 (λv) = (λ −1 λ)v = 1v = v v = 0<br />
F n n F<br />
X F X X → F F<br />
<br />
V F U ⊂ V V <br />
U ≤ V <br />
• 0 ∈ U<br />
• u1, u2 ∈ U =⇒ u1 + u2 ∈ U<br />
• λ ∈ F, u ∈ U =⇒ λu ∈ U<br />
U = ∅ U <br />
V F U ≤ V U F<br />
+ · V U<br />
R R R C(R) <br />
<br />
D(R) P (R) <br />
<br />
n ∈ N0 λ1, . . . , λn ∈ F v1, . . . , vn ∈ V n<br />
i=1 λivi <br />
λ1v1 + · · · + λnvn 0<br />
i=1 λivi = 0 <br />
S ⊂ V <br />
v∈S λvv <br />
v λv = 0<br />
v1, . . . , vn V V F v ∈ V <br />
v1, . . . , vn V = 〈v1, . . . , vn〉<br />
S ⊂ V S V <br />
∀v ∈ V ∃n ∈ N0 ∃v1, . . . , vn ∈ V ∃λ1, . . . , λn ∈ F v =<br />
P2(R) 1, x, x 2 <br />
P (R) <br />
n<br />
λivi.<br />
v1, . . . , vn V F <br />
λ1v1 + · · · λnvn = 0 λ1 = · · · = λn = 0 <br />
<br />
S ⊂ V S <br />
S <br />
0 1 · 0 = 0<br />
V = C R 1, i <br />
i=1
V = C C 1, i <br />
v1, . . . , vn V V F <br />
<br />
P2(R) 1, x, x 2 <br />
F n e1, . . . , en ei = (0, . . . , 1, . . . , 0) T <br />
{0} ∅<br />
v1, . . . , vn ∈ V V F v ∈ V<br />
v = n<br />
i=1 λivi λi ∈ F<br />
v ∈ V v1, . . . , vn V v = n i=1 λivi λi ∈ F <br />
v = n i=1 µivi 0 = n i=1 (λi − µi)vi v1, . . . , vn <br />
λi = µi i = 1, . . . , n<br />
v ∈ V v1, . . . , vn v1, . . . , vn<br />
V F n i=1 λivi = 0 = n i=1 0vi <br />
λi = 0 i = 1, . . . , n <br />
v1, . . . , vn V F {v1, . . . , vn} V <br />
v1, . . . , vn k <br />
α1, . . . , αk−1 ∈ F vk = α1v1 + · · · + αk−1vk−1 λ1v1 + · · · + λnvn = 0<br />
λi = 0 k λk = 0 αi = − λi <br />
λk<br />
v1, . . . , vk−1, vk+1, . . . , vn V v = n i=1 λivi v = k−1 i=1 (αi +<br />
λi)vi + n i=k+1 λivi <br />
V <br />
F v1, . . . , vm w1, . . . , wn V F <br />
m ≤ n wi v1, . . . , vm, wm+1, . . . , wn V <br />
r ≥ 0 wi wi <br />
v1, . . . , vr, wr+1, . . . , wn V r = m r < m <br />
vr+1 =<br />
r<br />
αivi +<br />
i=1<br />
n<br />
i=r+1<br />
αi, βi ∈ F βi = 0 i v1, . . . , vr, vr+1 <br />
wr+1, . . . , wn βr+1 = 0 <br />
wr+1 =<br />
r −αi<br />
βr+1<br />
i=1<br />
βiwi<br />
vi + 1<br />
vr+1 +<br />
βr+1<br />
<br />
i=r+2<br />
−βi<br />
wi.<br />
βr+1<br />
V v1, . . . , vr, vr+1, wr+2, . . . , wn m <br />
m wi vi <br />
m ≤ n<br />
V F <br />
V F dimF V
v1, . . . , vm w1, . . . , wn m ≤ n vi <br />
wi V n ≤ m wi <br />
vi V <br />
dimF F n = n<br />
dimR P2(R) = 3<br />
dimR C = 2<br />
dimF F = 1<br />
V F v1, . . . , vk<br />
k ≥ 0 v1, . . . , vk, vk+1, . . . , vn <br />
V <br />
v1, . . . , vk V vk+1 ∈ V \ 〈v1, . . . , vk〉 <br />
v1, . . . , vk+1 dim V − k <br />
V U ≤ V dim U ≤ dim V <br />
U = V <br />
V F dim V = n<br />
n <br />
n <br />
<br />
<br />
dimF V = n <br />
v1, . . . , vn <br />
v1, . . . , vn <br />
v1, . . . , vn V <br />
S ⊂ V U V <br />
S U = 〈S〉 S U <br />
S<br />
U S U <br />
V U <br />
S<br />
V = R R S = {1, x, x 2 , . . . } 〈S〉 = P (R) <br />
<br />
<br />
U, W ≤ V U + W = {u + w : u ∈ U, w ∈ W } U + W ≤ V
U ∪ W U ⊂ W W ⊂ U<br />
U W V U +W <br />
dim U + W = dim U + dim W − dim U ∩ W <br />
v1, . . . , vk U ∩ W v1, . . . , vk, u1, . . . , ul<br />
U v1, . . . , vk, w1, . . . , wm W <br />
v1, . . . , vk, u1, . . . , ul, w1, . . . , wm U + W <br />
• v ∈ U + W v = u + w u ∈ U w ∈ W u = αivi + βiui<br />
αi, βi ∈ F w = α ′ i vi + γiwi α ′ i , γi ∈ F <br />
v = (αi + α ′ i)vi + βiui + γiwi.<br />
• αivi + βiui + γiwi = 0 <br />
αivi + βiui = − γiwi<br />
= δivi<br />
δi ∈ F U W <br />
U ∩ W <br />
(αi − δi)vi + βiui = 0,<br />
v1, . . . , vk, u1, . . . , ul U βi 0 <br />
αivi + γiwi = 0,<br />
v1, . . . , vk, w1, . . . , wm W αi, γi 0<br />
V F U, W ≤ V <br />
V = U ⊕ W<br />
v V v = u + w u ∈ U w ∈ W <br />
W U V <br />
U, W ≤ V V = U ⊕ W U + W = V <br />
U ∩ W = {0}<br />
V F U ≤ V U <br />
V U = {0} U = V <br />
v1, . . . , vk U u1, . . . , uk, wk+1, . . . , wn <br />
V W = 〈wk+1, . . . , wn〉 U V <br />
V1, . . . , Vk ≤ V Vi = { vi : vi ∈ Vi} ≤ V <br />
Vi v ∈ V vi vi ∈ Vi<br />
V1, . . . , Vk ≤ V <br />
Vi
Bi Vi B = k<br />
i=1 Bi Vi<br />
i Vi ∩ <br />
j=i Vj = {0}<br />
k > 2 Vi ∩ Vj = {0} i = j<br />
=⇒ Bi Vi B = k i=1 Bi v ∈ Vi<br />
v = k i=1 vi vi Bi <br />
vi v B <br />
B 0 Vi vi <br />
Vi v1 +· · ·+vk = 0 0 vi = 0<br />
Bi 0
V W F α : V → W <br />
v, v1, v2 ∈ V λ ∈ F<br />
α(v1 + v2) = α(v1) + α(v2)<br />
α(λv) = λα(v)<br />
D : D(R) → F (R) = R R f ↦→ f<br />
t<br />
x<br />
0 : C[0, 1] → F [0, 1] f ↦→ x<br />
0 f(t) t <br />
<br />
A m × n α : F n → F m x ↦→ Ax <br />
U, V, W <br />
ιv : V → V, v ↦→ v <br />
U β −→ V α −→ W α, β α ◦ β : U → W <br />
V W F B V α0 : B → W<br />
α : V → W α0 α(v) = α0(v)<br />
v ∈ B<br />
v ∈ V v = λ1v1 + · · · + λnvn vi ∈ B λi ∈ F <br />
α(v) = λiα0(vi) α <br />
V W F α : V → W <br />
V W V → W <br />
F<br />
ιV : V → V <br />
α : V → W α −1 : W → V <br />
U α −→ V β −→ W β ◦ α : U → W
α α −1 : W → V<br />
<br />
α −1 <br />
α −1 (w1 + w2) = α −1 (α(v1) + α(v2))<br />
= α −1 (α(v1 + v2))<br />
= v1 + v2<br />
= α −1 (w1) + α −1 (w2)<br />
α −1 (λw) = α −1 (λα(v))<br />
= α −1 (α(λv))<br />
= λv<br />
= λα −1 (w)<br />
V F n V F n <br />
v1, . . . , vn α : V → F n , n<br />
i=1 λivi ↦→ (λ1, . . . , λn) T <br />
<br />
V W F <br />
<br />
v1, . . . , vn w1, . . . , wn V W <br />
α : V → W, λivi ↦→ λiwi <br />
V W B <br />
V α : V → W α(B) W <br />
• w ∈ W w = α(v) v ∈ V v = n<br />
i=1 λivi v1, . . . , vn ∈ B<br />
λi ∈ F w = α( λivi) = λiα(vi)<br />
• λ1α(v1)+· · ·+λnα(vn) = 0 α(λ1v1+· · ·+λnvn) = 0 λ1v1+· · ·+λnvn = 0<br />
α λi = 0 i B <br />
<br />
α : V → W ker(α) = {v ∈ V : α(v) =<br />
0} = N(α) α Im(α) = {w ∈ W : w = α(v) v ∈ V }<br />
N(α) ≤ V Im(α) ≤ W α N(α) = {0} <br />
Im(α) = W n(α) = dim N(α) α <br />
rank(α) = dim Im(α) α<br />
V W F dimF V<br />
α : V → W dim V = rank(α) + n(α)<br />
v1, . . . , vk N(α) v1, . . . , vk, vk+1, . . . , vn <br />
V α(vk+1), . . . , α(vn) Im(α)<br />
• w ∈ Im(α) w = α(v) v ∈ V v = n i=1 λivi λi ∈ F <br />
w = α(v) = n i=1 λiα(vi) = n k+1 λiα(vi) α(vi) = 0 i = 1, . . . , k
• n i=k+1 λiα(vi) = 0 α( n i=k+1 λivi) = 0 n i=k+1 λivi ∈ N(α) <br />
k i=1 λivi v1, . . . , vn λi = 0 <br />
i = 1, . . . , n<br />
<br />
V F N ≤ V V/N = {v + N : v ∈ V } <br />
F <br />
(v1 + N) + (v2 + N) = (v1 + v2) + N<br />
λ(v + N) = (λv) + N<br />
¯ V = V/N ¯v = v + N v1, . . . , vk, vk+1, . . . , vn V <br />
v1, . . . , vk N(α) vk+1, ¯ . . . , vn ¯ ¯ V dim V/N =<br />
dim V − dim N α : V → W W F<br />
V/N(α) Im(α) v + N(α) ↦→ α(v) dim Im(α) =<br />
dim V/N(α) = dim V − dim N(α)<br />
V F α : V → V <br />
α : V → W dim V = dim W <br />
<br />
α <br />
α <br />
α <br />
<br />
<br />
U V F L(U, V ) = {a : U → V | α } <br />
<br />
(α1 + α1)(u) = α1(u) + α2(u)<br />
(λα)(u) = λα(u)<br />
u ∈ U α, α1, α2 ∈ L(U, V ) λ ∈ F F <br />
U → V L(U, V ) <br />
U V F L(U, V ) F<br />
U V L(U, V ) dim L(U, V ) = dim U dim V <br />
u1, . . . , un U <br />
v1, . . . , vm V 1 ≤ i ≤ m 1 ≤ j ≤ n εij : uk ↦→ δjkvi <br />
1 ≤ k ≤ n εij ∈ L(U, V ) <br />
<br />
i,j λijεij = 0 1 ≤ k ≤ n<br />
0 = <br />
λijεij(uk) = <br />
λikvi.<br />
i,j<br />
i
vi λik = 0 1 ≤ i ≤ m <br />
1 ≤ k ≤ n εij <br />
α ∈ L(U, V ) α(uk) = <br />
i aikvi <br />
<br />
(aijεij(uk)) = <br />
aikvi = α(uk)<br />
i,j<br />
1 ≤ k ≤ n U <br />
<br />
<br />
m × n F A = (aij) m n <br />
aij ∈ F 1 ≤ i ≤ m 1 ≤ n ≤ n Mm,n(F) m × n <br />
F<br />
Mm,n(F) <br />
dimF Mm,n(F) = mn<br />
(aij) + (bij) = (aij + bij)<br />
λ(aij) = (λaij)<br />
1 ≤ i ≤ m 1 ≤ j ≤ n <br />
<br />
eij = 1<br />
Eij =<br />
<br />
i<br />
ei ′ j ′ = 0 (i′ , j ′ ) = (i, j).<br />
U V F α : U → V <br />
B = {u1, . . . , un} C = {v1, . . . , vm} U V A = (aij) <br />
α(uj) = <br />
i aijvi u ∈ U u = <br />
i λiui [u]B = (λ1, . . . , λn) T <br />
A = <br />
[α(u1)]C · · · [α(un)]C ,<br />
A = [α]B,C<br />
u ∈ U [α(u)]C = [α]B,C[u]B<br />
u ∈ U u = <br />
j λjuj [u]B = (λ1, . . . , λn) T <br />
α(u) = <br />
λjα(uj)<br />
[α(u)]C = A · [u]B<br />
j<br />
= <br />
j<br />
λj<br />
= <br />
i<br />
j<br />
<br />
i<br />
aijvi<br />
aijλjvi<br />
= <br />
(A · [u]B)ivi<br />
i
B = {u1, . . . , un} C = {v1, . . . , vm} U, V <br />
ε = εB : U → F n , u ↦→ [u]B φ = φC : V → F m , v ↦→ [v]C <br />
<br />
εB<br />
U<br />
⏐<br />
<br />
F n<br />
α<br />
−−−−→ V<br />
⏐<br />
⏐<br />
φC<br />
A·<br />
−−−−→ F m<br />
A [α(u)]C = A · [u]B<br />
u ∈ U A ′ · [u]B = [α(u)]C u ∈ U <br />
u1, . . . , un [uk]B = ek A ′ · ek k A ′ <br />
1 ≤ k ≤ n <br />
α : U → V dim U = n dim V = m L(U, V ) <br />
Mm,n(F)<br />
B = {u1, . . . , un} C = {v1, . . . , vm} U V <br />
θ : L(U, V ) → Mm,n(F), α ↦→ [α]B,C <br />
U α −→ V β −→ W B, C, D U, V, W <br />
[β ◦ α]B,D = [β]C,D · [α]B,C<br />
A = [α]B,C B = [β]C,D <br />
[β ◦ α]B,D = BA<br />
β ◦ α(uk) = β <br />
= <br />
j<br />
j<br />
ajk<br />
= <br />
i<br />
j<br />
ajkvj<br />
<br />
i<br />
bijwi<br />
bijajkwi<br />
= <br />
(BA)ikwi<br />
i<br />
<br />
U V F U B = {u1, . . . , un} <br />
B ′ = {u ′ 1 , . . . , u′ 2 } V C = {v1, . . . , vm} C ′ = {v ′ 1 , . . . , v′ m} <br />
P = (pij) B B ′ u ′ <br />
j = i pijui <br />
P = [u ′ 1 ]B . . . [u ′ <br />
n]B = [ιU]B ′ B.<br />
[u]B = P [u]B ′ u ∈ U u′ j [u′ j ]B ′ = ej <br />
P P −1 B ′ B <br />
Q C C ′ <br />
α : U → V B, B ′ , C, C ′ <br />
A = [α]B,C A ′ = [α]B ′ ,C ′ A′ = Q −1 AP
u ∈ U <br />
A ′ = Q −1 AP <br />
[α(u)]C = A[u]B = AP [u]B ′<br />
= Q[α(u)]C ′<br />
m × n A, A ′ ∈ Mm,n(F) <br />
Q ∈ Mm,m(F) P ∈ Mn,n(F) A ′ = QAP <br />
Mm,n(F)<br />
U <br />
V m n <br />
U V F dim U = n dim V = m<br />
α : U →<br />
<br />
V <br />
r<br />
B U C V <br />
[α]B,C = Ir 0<br />
0 0<br />
m × n Ir 0<br />
0 0<br />
r<br />
ur+1, . . . , un N(α) B =<br />
{u1, . . . , ur, ur+1, . . . , un} U α(u1), . . . , α(r) Im(α) <br />
<br />
<br />
C = {α(u1), . . . , α(ur), vr+1, . . . , vm} V <br />
<br />
[α]B,C = Ir 0<br />
0 0<br />
A ∈ Mm,n(F) α : Fn → Fm , x ↦→ Ax <br />
Fn Fm <br />
α A A <br />
Ir 0<br />
0 0 r<br />
<br />
A ∈ Mm,n(F) A rank(A) <br />
A F n <br />
A<br />
α : U → V B U C V <br />
A = [α]B,C rank(α) = rank(A)<br />
θ : Im(α) → colsp(A), α(u) ↦→ [α(u)]C <br />
A, A ′ ∈ Mm,n(F) <br />
rank(A) = rank(A ′ )<br />
A ′ = Q −1 AP Q, P α : F n → F m , x ↦→ Ax <br />
α B, C A B ′ <br />
P C ′ Q ′ [α]B ′ ,C ′ = Q−1 AP <br />
P C B B ′ C C ′ <br />
rank(A) = rank(α) = rank(A ′ ) <br />
A A ′ <br />
Ir 0 Ir ′ 0<br />
r r ′ <br />
0 0 0 0<br />
rank(A) = r rank(A ′ ) = r ′ rank(A) = rank(A ′ ) r = r ′<br />
A A ′
A ∈ Mm,n(F) rowrk(A) = dim rowsp(A) = rank(A T )<br />
A ∈ Mm,n(F) rowrk(A) = rank(A)<br />
A ∈ Mm,n(F) r = rank(A) A <br />
Ir 0<br />
0 0 <br />
<br />
m×n<br />
Ir 0<br />
0 0 = QAP Q P <br />
m×n<br />
P T A T Q T <br />
Ir 0<br />
=<br />
0 0<br />
n×m<br />
AT <br />
Ir 0<br />
0 0 n×m rank <br />
Ir 0<br />
0 0 = r rowrk(A) =<br />
n×m<br />
rank(AT ) = rank <br />
Ir 0<br />
0 0 = r = rank(A)<br />
n×m<br />
<br />
m × n <br />
F<br />
i j<br />
i λ i λ ∈ F \ {0}<br />
λ i j i = j λ ∈ F<br />
In<br />
Tij Mi,λ Ci,j,λ A <br />
<br />
<br />
m × n <br />
<br />
j 1 ij i1 ≤ i2 ≤ · · · <br />
ij k k < j 0<br />
A <br />
<br />
A n × n <br />
In A −1 <br />
A ↦→ AE1E2 · · · Ek = I<br />
In ↦→ InE1E2 · · · Ek = A −1 .<br />
A n × n A <br />
<br />
A −1 = E1 · · · Ek <br />
A = E −1<br />
k · · · E −1<br />
1 <br />
n × n A A ′ A ′ = P −1 AP <br />
P
A ∈ Mn(F) tr A = n<br />
i=1 aii tr : Mn(F) → F <br />
tr(AB) = tr(BA)<br />
tr(AB) = <br />
i j aijbji = <br />
j i bjiaij = tr(BA)<br />
<br />
tr(P −1 AP ) = tr(AP P −1 ) = tr(A)<br />
α ∈ End(V ) tr(α) = tr[α]B B V <br />
Sn {1, . . . , n} <br />
(σ ◦ τ)(j) = σ(τ(j)) σ<br />
<br />
<br />
+1 <br />
ε(σ) =<br />
−1 <br />
ε : Sn → {+1, −1} <br />
A ∈ Mn(F) <br />
det A = <br />
σ∈Sn<br />
ε(σ)a σ(1)1 · · · a σ(n)n.<br />
A (i) i A A = (A (1) , . . . , A (n) ) <br />
A n F n {e1, . . . , en} <br />
F n <br />
d : F n × · · · × F n → F F n <br />
<br />
d(v1, . . . , λivi, . . . , vn) = λd(v1, . . . , vi, . . . , vn)<br />
d(v1, . . . , vi + v ′ i, . . . , vn) = d(v1, . . . , vi, . . . , vn) + d(v1, . . . , v ′ i, . . . , vn)<br />
i = j vi = vj d(v1, . . . , vn) = 0
d d(e1, . . . , en) = 1<br />
i = j<br />
<br />
d(v1, . . . , vj, . . . , vi, . . . , vn) = −d(v1, . . . , vi, . . . , vj, . . . , vn).<br />
0 = d(v1, . . . , vi + vj, . . . , vi + vj, . . . , vn)<br />
= 0 + d(v1, . . . , vi, . . . , vj, . . . , vn) + d(v1, . . . , vj, . . . , vi, . . . , vn) + 0<br />
σ ∈ Sn d F n <br />
v1, . . . , vn ∈ F n <br />
d <br />
d(v σ(1), . . . , v σ(n) = ε(σ)d(v1, . . . , vn)<br />
d(e σ(1), . . . , e σ(n) = ε(σ)d(e1, . . . , en)<br />
= ε(σ)<br />
d F n A = (aij) = (A (1) , . . . , A (n) ) ∈ Mn(F)<br />
d(A (1) , . . . , A (n) ) = det Ad(e1, . . . , en)<br />
<br />
d(A (1) , . . . , A (n) ) = d( <br />
j1<br />
aj11ej1 , A(2) , . . . , A (n) )<br />
= <br />
aj11d(ej1 , A(2) , . . . , A (n) )<br />
j1<br />
= <br />
aj11aj22d(ej1 , ej2 , A(3) , . . . , A (n) )<br />
j1,j2<br />
= · · ·<br />
= <br />
j1,...,jn<br />
aj11 · · · ajnnd(ej1 , . . . , ejn)<br />
= <br />
aσ(1)1 · · · aσ(n)nε(σ)d(e1, . . . , en)<br />
σ∈Sn<br />
= (det A)d(e1, . . . , en).<br />
d : F n × · · · × F n → F d(A (1) , . . . , A (n) ) = det A <br />
A = (A (1) , . . . , A (n) ) d <br />
n<br />
j=1 a σ(j)j det A<br />
A (k) = A (l) k = l det A = 0 τ = (kl) Sn<br />
det A = <br />
ε(σ) <br />
aσ(j)j σ∈Sn<br />
j
σ <br />
στ ε(σ) = 1 ε(στ) = −1 <br />
det A = <br />
⎛<br />
⎝ <br />
aσ(j)j − <br />
⎞<br />
a ⎠<br />
στ(j)j = 0<br />
σ <br />
<br />
det A = <br />
σ∈Sn ε(σ) n<br />
det A T = det A<br />
j<br />
a σ(1)1 · · · a σ(k)k · · · a σ(l)l · · · a σ(n)n<br />
− a σ(1)1 · · · a σ(l)k · · · a σ(k)l · · · a σ(n)n = 0.<br />
j=1 δ σ(j)j = ε(ι) · 1 = 1<br />
σ ∈ Sn n<br />
j=1 a σ(j)j = n<br />
j=1 a jσ(j) <br />
σ Sn σ −1 ε(σ −1 ) = ε(σ) <br />
det A = <br />
ε(σ)<br />
σ∈Sn<br />
= <br />
ε(σ)<br />
σ∈Sn<br />
= <br />
ε(σ)<br />
σ∈Sn<br />
= det A T<br />
j<br />
n<br />
aσ(j)j j=1<br />
n<br />
ajσ−1 (j)<br />
j=1<br />
n<br />
ajσ(j) det I<br />
A aij = 0 i > j det A =<br />
a11 · · · ann<br />
<br />
det A = <br />
σ∈Sn<br />
j=1<br />
ε(σ)a σ(1)1 · · · a σ(n)n.<br />
σ(i) ≤ i i = 1, . . . , n σ(1) = 1<br />
σ(2) = 2 σ(n) = n σ = ι det A = a11 · · · ann<br />
E n × n A<br />
det(AE) = det A det E = det(EA).<br />
A det A <br />
<br />
det Tij = −1<br />
det Mi,λ = λ det Ci,j,λ = 1 <br />
det A −1 λ 1
A A <br />
det A = 0<br />
A A <br />
det A det A = 0<br />
A 0 A<br />
A 0 <br />
det A = 0 <br />
A, B ∈ Mn(F) det(AB) = det(A) det(B)<br />
A dA : (B (1) , . . . , B (n) ) ↦→ det(AB) B = (B (1) , . . . , B (n) ) <br />
F n det(AB) = dA(AB (1) , . . . , AB (n) ) dA <br />
<br />
<br />
det(AB) = dA(B (1) , . . . , B (n) )<br />
= det BdA(e1, . . . , en)<br />
= det(B) det(A),<br />
<br />
⎛<br />
det(AB) = det ⎝ <br />
bj11A (j1)<br />
<br />
, . . . , bjnnA (jn)<br />
⎞<br />
⎠<br />
σ∈Sn<br />
j1<br />
= <br />
⎛ ⎞<br />
n<br />
⎝ b ⎠<br />
σ(j)j det(A σ(1) , . . . , A σ(n) )<br />
σ∈Sn<br />
j=1<br />
jn<br />
= <br />
⎛ ⎞<br />
n<br />
⎝ b ⎠<br />
σ(j)j ε(σ) det A<br />
j=1<br />
= det(A) det(B).<br />
B AB det B = 0 = det(AB) B <br />
B = E1 · · · Ek <br />
<br />
det(AB) = det(AE1 · · · Ek)<br />
= det A det E1 · · · det Ek<br />
= det A det B.<br />
A det A −1 = (det A) −1 <br />
A AA −1 = I (det A)(det A −1 ) = det I = 1 <br />
det A −1 = (det A) −1 <br />
n × n
det(P −1 AP ) = det P −1 det A det P<br />
= (det A)(det P )(det P ) −1<br />
= det A<br />
α : V → V det α = det[α]B <br />
B V <br />
det : End(V ) → F <br />
det ι = 1<br />
det α ◦ β = det α det β<br />
det α = 0 α α det α −1 =<br />
(det α) −1 <br />
GL(V ) V GLn(F) <br />
n × n F GL(V ) GLn(F) det : GLn(F) → F <br />
<br />
A ∈ Mm(F) B ∈ Mk(F) C ∈ Mm,k(F) det <br />
A C<br />
0 B =<br />
det A det B<br />
B, C dB,C : A ↦→ det <br />
A C<br />
0 B <br />
Fm dB,C(A) = det A det <br />
I C<br />
0 B C <br />
B ↦→ det <br />
I C<br />
0 B Fk det I C<br />
0 B = det B I C<br />
0 I <br />
det <br />
I C<br />
0 I = 1 I C<br />
0 I det A C<br />
0 B = det A det B<br />
X = <br />
A C<br />
0 B <br />
<br />
A C<br />
det =<br />
0 B<br />
m+n <br />
ε(σ) xσ(j)j σ∈§m+n j=1<br />
x σ(j)j = 0 j ≤ m σ(j) > m σ <br />
j ∈ [1, m] σ(j) ∈ [1, m] x σ(j)j = a σ1(j)j σ1 ∈ Sm <br />
σ [1, m]<br />
j ∈ [m + 1, m + k] σ(j) ∈ [m + 1, m + k] l = j − m <br />
x σ(j)j = b σ2(l)l σ2(l) = σ(m + l) − m<br />
ε(σ) = ε(σ1)ε(σ2) σ <br />
det ⎛<br />
<br />
A C<br />
0 B = ⎝ <br />
⎞ ⎛<br />
m<br />
ε(σ1) a ⎠ ⎝<br />
σ1(j)j<br />
<br />
⎞<br />
k<br />
ε(σ2) a ⎠<br />
σ2(j)j<br />
σ1∈Sm<br />
= det A det B<br />
j=1<br />
σ2∈Sk<br />
A = (aij) n × n A ij (n − 1) × (n − 1)<br />
A i j<br />
l=1
j det A = n<br />
i=1 (−1)i+j aij det(A ij )<br />
i det A = n<br />
j=1 (−1)i+j aij det(A ij )<br />
<br />
<br />
det A = d(A (1) , . . . , A (n) )<br />
=<br />
=<br />
n<br />
aij(−1) i+j−2 <br />
1 ∗<br />
d<br />
0 A ij<br />
i=1<br />
n<br />
i=1<br />
(−1) i+j aij det A ij ,<br />
<br />
A ∈ Mn(F) adj A n × n (i, j) <br />
(−1) i+j det A ij <br />
(adj A)A = (det A)I<br />
A A−1 = 1<br />
det A adj A<br />
<br />
j < k<br />
det A =<br />
n<br />
(adj A)jiaij = (adj A · A)jj.<br />
i=1<br />
0 = det(A (1) , . . . , A (k) , . . . , A (k) , . . . , A (n) )<br />
=<br />
n<br />
(adj A)jiaik<br />
i=1<br />
= (adj A · A)jk.<br />
1<br />
A det A = 0 <br />
adj A<br />
A −1 = 1<br />
det A<br />
det A<br />
adj A · A = I <br />
Ax = b m n <br />
A m × n b F m rank A =<br />
rank(A| b) n = rank A = rank(A| b) <br />
x = A −1 b <br />
m = n <br />
A ∈ Mn(F) Ax = b x =<br />
(x1, . . . , xn) T xi = 1<br />
detA det A îb i = 1, . . . , n A îb <br />
A i b
x Ax = b <br />
det A îb = det(A(1) , . . . , A (i−1) , b, A (i+1) , . . . , A (n) )<br />
= <br />
j=1<br />
= xi det A,<br />
xi = 1<br />
detA det Aîb i = 1, . . . , n<br />
xj det(A (1) , . . . , A (i−1) , A (j) , A (i+1) , . . . , A (n) )<br />
A ∈ Mn(Z) det A = ±1 b ∈ Z n Ax = b<br />
Z
V F F<br />
R C α : V → V <br />
α ∈ End(V ) α B V <br />
[α]B i = j aij = 0 α B<br />
V [α]B i > j aij = 0<br />
<br />
<br />
α ∈ End(V ) λ ∈ F α <br />
v ∈ V v = 0 α(v) = λv v λ<br />
λ α α − λι det(α − λι) = 0<br />
λ = 0 α <br />
χα(t) = det(α − tι) α<br />
n = dim V A ∈ Mn(F) χα(t) = det(A − tI)<br />
α <br />
A λ A χA(λ) = 0 v ∈ F n <br />
v = 0 Av = λv<br />
<br />
<br />
<br />
χ P −1 AP (t) = det(P −1 AP − tI)<br />
= det P −1 det(A − tI) det P<br />
= χA(t).<br />
<br />
1 1<br />
0 1 2 × 2 <br />
1, 1 I <br />
1 C <br />
n <br />
V C α ∈ End(V ) α<br />
V
V C α ∈ End(V )<br />
B V [α]B <br />
B = {v1, . . . , vn} α(vj) ∈ 〈v1, . . . , vj〉 j = 1, . . . , n<br />
n n = 1 n > 1 <br />
V C λ ∈ C α − λι <br />
U = Im(α−λι) V U <br />
α(U) ⊂ U<br />
α(U) = α((α − λι)(V )) = (α − λι)(αV ) ≤ (α − λι)(V ) = U.<br />
α ′ = α| U : U → U dim U < dim V <br />
B ′ = {v1, . . . , vk} U A ′ = [α ′ ]B ′ <br />
B = {v1, . . . , vk, . . . , vn} V [α]B <br />
[α]B =<br />
<br />
A ′ ∗<br />
.<br />
0 λI<br />
1 ≤ j ≤ k α(vj) = α ′ (vj) ∈ U k <br />
j > k (α − λι)(vj) ∈ U U α(vj) = λvj + u u ∈ U <br />
k j u B ′ <br />
λ (j, j)<br />
v1 α α(v1) = λv1 U <br />
〈v1〉 V v ∈ V v = λvv1+u u ∈ U λv ∈ F <br />
π(v) = u V U u ∈ U ˜α : U → U ˜α(u) = π(α(u))<br />
˜α ∈ End(U) v2, . . . , vn U ˜α(vj) ∈ 〈v2, . . . , vj〉<br />
2 ≤ j ≤ n α(vj) = λ α(vj)vj + ˜α(vj) ∈ 〈v1, . . . , vj〉 α(v1) ∈ 〈v1〉<br />
C <br />
R ±I R 2 <br />
V F α ∈<br />
End(V ) B V [α]B <br />
χα χα F<br />
<br />
[α]B =<br />
⎛<br />
⎜<br />
⎝<br />
a11<br />
<br />
∗<br />
0 ann<br />
χα(t) = (a11 − t) · · · (ann − t) aij ∈ F<br />
dimF V λ F U =<br />
(α − λι)(V ) α(U) ≤ U V α ′ = α| U ∈ End(U) B ′ U<br />
B V <br />
[α]B =<br />
⎞<br />
⎟<br />
⎠<br />
<br />
[α ′<br />
]B ′ ∗<br />
.<br />
0 λI<br />
χα(t) = χα ′(t)χλI(t) χα ′ F <br />
B ′ [α ′ ]B ′
α ∈ End(V ) U V α(U) ≤ U<br />
B ′ = {v1, . . . , vk} U B = {v1, . . . , vk, . . . , vn}<br />
V ¯ V = V/U ¯v = v + U v ∈ V ¯ B = {¯vk+1, . . . , ¯vn} ¯ V <br />
α ′ = α| U ∈ End(U) ¯α : ¯ V → ¯ V , ¯v ↦→ ¯ α(v) <br />
¯ V <br />
<br />
[α ′<br />
]B ′ ∗<br />
[α]B =<br />
.<br />
0 [¯α] B¯<br />
χα = χα ′ · χ¯α<br />
V F α ∈ End(V ) B <br />
V [α]B B α<br />
α ∈ End(V ) λ1, . . . , λk α Vj =<br />
N(α−λjι) λj V1 +· · ·+Vk Bj <br />
Vj k j=1 Bj V1+· · ·+Vk k j=1 dim Vj = dim V<br />
[α]B V = V1 ⊕ · · · ⊕ Vk<br />
v1+. . .+vk = 0 vj ∈ Vj vj = 0 j = 1, . . . , k<br />
<br />
v1 + · · · + vj = 0<br />
α λ1 <br />
<br />
α(v1) + · · · + α(vj) − λ1v1 − · · · − λ1vj = 0<br />
⇐⇒ (λ2 − λ1)v2 · · · + (λj − λ1)vj = 0,<br />
Vj = Vj <br />
<br />
V F α ∈ End(V )<br />
F <br />
<br />
[α]B =<br />
⎛<br />
⎜<br />
⎝<br />
λ1<br />
<br />
0 λk<br />
⎞<br />
0<br />
⎟<br />
⎠<br />
λ1, . . . , λk p(t) = k<br />
j=1 (λj − t) v ∈ B α(v) = λlv <br />
l ≤ k (λlι − α)v = 0 p(α)(v) = 0 p(α) 0 <br />
B<br />
v ∈ V <br />
p(t) = k<br />
j=1 (λj − t) λ1, . . . , λk <br />
pj(t) = (λ1 − t) · · · (λj−1 − t)(λj+1 − t) · · · (λk − t)<br />
hj(t) = pj(t)<br />
pj(λj)
hj(λi) = δij 1 ≤ i, j ≤ k h(t) = k<br />
j=1 hj(t) = 1 h(t) − 1 <br />
k λ1, . . . , λk k v ∈ V <br />
v = ι(v) = h(α)(v) =<br />
k<br />
hj(α)(v) =<br />
vj = hj(α)(v) (α − λjι)vj = 0 p(α) = 0 vj <br />
α λj v ∈ V <br />
hj(α) <br />
<br />
A ∈ Mn(F) P −1AP P p(A) = 0<br />
p ∈ F[t] P <br />
P −1 ⎛<br />
d1<br />
⎜<br />
AP = D = ⎝<br />
⎞<br />
0<br />
⎟<br />
⎠<br />
0 dn<br />
AP = P D<br />
AP (j) = djP (j)<br />
j P A dj<br />
α1, α2 ∈ End(V ) <br />
α1α2 = α2α1 <br />
B V [α1]B [α2]B <br />
V = V1⊕· · ·⊕Vk Vj α1 α1(vj) = λjvj<br />
vj ∈ Vj α2(Vj) ⊂ Vj v ∈ Vj α1(α2(v)) = α2(α1(v)) = α2(λjv) = λjα2(v)<br />
α2(v) ∈ Vj α2| Vj Bj <br />
Vj α2 α1 <br />
B α1 α2<br />
<br />
<br />
p(t) ∈ F[t]<br />
j=1<br />
p(t) = ant n + · · · + a1t + a0<br />
ai ∈ F 0 ≤ i ≤ n p(t), q(t) ∈ F[t] F[t] <br />
<br />
m ≤ n<br />
p(t) = ant n + · · · + a1t + a0<br />
q(t) = bmt m + · · · + b1t + b0<br />
(p + q)(t) = ant n + · · · + (am + bm)t m + · · · + (a1 + b1)t + (a0 + b0)<br />
k<br />
j=1<br />
(pq)(t) = anbmt n+m + · · · + (a1b0 + a0b1)t + a0b0<br />
vj
deg p p l al = 0 −∞ p <br />
0 deg pq = deg p + deg q<br />
F[t] a, b ∈ F[t] b = 0 <br />
q, r ∈ F[t] a = bq + r deg r < deg b r = 0 a = antn + · · · + a0<br />
b = bmtm + · · · + b0 bm = 0 n ≥ m q = 0<br />
r = a a a ′ = a − an<br />
bm tn−mb deg a ′ < deg a a ′ = bq ′ + r<br />
q ′ , r deg r < deg b q = an<br />
bm tn−mq ′ <br />
F[t] <br />
p ∈ F[t] λ ∈ F p(λ) = 0 q ∈ F[t] p(t) = (λ − t)q(t)<br />
λ p e (λ − t) e p (λ − t) e+1 <br />
n n <br />
p1, p2 n n <br />
p, q ∈ F[t] α ∈ End(V ) p(α)q(α) = q(α)p(α)<br />
α(v) = λv p ∈ F[t] p(α)(v) = p(λ)v<br />
V F dim V = n <br />
α ∈ End(V ) p n 2 p(α) = 0<br />
dim End(V ) = n 2 a n 2, . . . , a1, a0 ∈ F <br />
an2α n2<br />
+ an2−1α n2−1 + · · · + a1α + a0ι = 0<br />
n2 +1 p(t) = an2tn2 +· · ·+a1t+a0<br />
α ∈ End(V ) mα α <br />
mα(α) = 0<br />
α ∈ End(V ) p ∈ F[t] p(α) = 0 mα p<br />
F[t] p = mαq + r q, r ∈ F[t] <br />
deg r < deg mα r = 0 p(α) = 0 = mα(α) r(α) = 0 r = 0 <br />
deg mα<br />
<br />
V <br />
α ∈ End(V ) χα(α) = 0<br />
A ∈ Mn(F) χA(A) = 0<br />
A ∈ Mn(F) <br />
(−1) n χA(t) = t n + an−1t n−1 + · · · + a1t + a0 = det(tI − A)<br />
B B · adj B = (det B)I adj(tI − A) <br />
n<br />
(tI − A)(Bn−1t n−1 + · · · + B1t + B0) = (tI − A) adj(tI − A)<br />
= (t n + an−1t n−1 + · · · + a1t + a0)I
I = Bn−1<br />
an−1I = Bn−2 − ABn−1<br />
<br />
a1I = B0 − AB1<br />
a0I = −AB0<br />
j A n+1−j <br />
<br />
A n + an−1A n−1 + · · · a1A + a0I = 0,<br />
= It n + an−1It n−1 + · · · + a1It + a0I<br />
C α ∈ End(V ) B = {v1, . . . , vn} V α(vj) ∈<br />
〈v1, . . . , vj〉 = Uj <br />
⎛<br />
λ1<br />
⎜<br />
[α]B = ⎝<br />
⎞<br />
∗<br />
⎟<br />
⎠<br />
0 λn<br />
(α − λjι)Uj ⊂ Uj−i <br />
χα(t) = (λ1 − t) · · · (λn − t)<br />
(α − λ1ι) · · · (α − λn−1ι)(α − λnι)V<br />
⊂(α − λ1ι) · · · (α − λn−1ι)Un−1<br />
⊂ · · ·<br />
mα χα<br />
⊂(α − λ1ι)U1 = 0.<br />
V C dim V = n <br />
χα(t) =<br />
k<br />
(t − λj) aj<br />
j=1<br />
λ1, . . . , λk α aj λj<br />
k<br />
j=1 aj = n<br />
mα(t) = k<br />
j=1 (t − λj) ej ej 1 ≤ ej ≤ aj 1 ≤ j ≤ k<br />
mα χα ej ≤ aj 1 ≤ j ≤ k λ α(v) = λv <br />
v = 0 0 = mα(α)v = mα(λ)v v = 0 mα(λ) = 0 (t − λ) <br />
mα(t)<br />
V F α ∈ End(V )<br />
α λ1, . . . , λk α <br />
mα(t) = k j=1 (t − λj)
α p(α) = 0 <br />
mα <br />
mα(t) = k j=1 (t − λj)<br />
V a ∈ End(V ) <br />
λ1, . . . , λk α λj N(α−λjι) <br />
λj gj = dim N(α − λjι) 1 ≤ gj ≤ aj α<br />
aj = gj 1 ≤ j ≤ k<br />
λj 1 ≤ gj B v1, . . . , vgj<br />
N(α − λjι) <br />
<br />
λjIgj [α]B =<br />
0<br />
∗<br />
A ′<br />
<br />
χα(t) = (λj − t) gj χA ′(t) gj ≤ aj<br />
χA(t) = (−1) n t n + an−1t n−1 + · · · + a0 a0 = det A an−1 =<br />
(−1) n−1 tr A<br />
V C End(V ) <br />
<br />
<br />
<br />
0 1 <br />
0 <br />
J(s, λ) <br />
⎛<br />
λ<br />
⎜<br />
J(s, λ) = ⎜<br />
⎝<br />
1<br />
⎞<br />
0<br />
⎟<br />
1⎠<br />
0 λ<br />
J(s, λ) (λ − t) s <br />
(λ − t) s λ 1<br />
V C α ∈ End(V ) <br />
B A = [α]B <br />
A =<br />
⎛<br />
⎜<br />
⎝<br />
B1<br />
<br />
0 Bk<br />
⎞<br />
0<br />
⎟<br />
⎠<br />
aj × aj Bj λj 1 ≤ j ≤ k λ = λj<br />
Bj <br />
Bj =<br />
⎛<br />
⎜<br />
⎝<br />
C1<br />
<br />
0<br />
0 Cm<br />
⎞<br />
⎟<br />
⎠<br />
s×s
m = mj Cl = J(nl, λ) <br />
aj<br />
aj = mj<br />
i=1 ni gj = mj ej = n1<br />
n1 ≥ n1 ≥ · · · nm > 0<br />
n1 + n2 + · · · + nm = 0<br />
C <br />
λ1, . . . , λk<br />
<br />
V Wj = N(α − λjι) aj V =<br />
k j=1 Wj B = k j=1 Bj Bj Wj <br />
⎛<br />
B1<br />
⎜<br />
[α]B = ⎝<br />
⎞<br />
0<br />
⎟<br />
⎠ .<br />
0 Bk<br />
pj(t) = (λj −t) −aj k r=1 (λr −t) ar qj k Wj = Im(hj(α))<br />
j=1 pjqj = 1<br />
α(Wj) ⊂ Wj Wj α| ∈ End(Wj)<br />
Wj<br />
λ = λj V = Wj n = aj (α − λι) n = 0 α − λι <br />
V <br />
⎛<br />
λ<br />
⎜<br />
⎜1<br />
⎜<br />
⎝<br />
⎞<br />
0<br />
⎟<br />
⎠<br />
0 1 λ<br />
.<br />
λ v1, . . . , vm <br />
α−λι v1 ↦→ v2 ↦→ . . . ↦→ vm ↦→ 0 <br />
<br />
n = 3 <br />
<br />
⎛ ⎞<br />
⎛ ⎞ ⎛ ⎞<br />
⎝<br />
λ1<br />
λ2<br />
λ3<br />
⎠<br />
⎝<br />
λ1<br />
λ2 1 ⎠<br />
(λ1 − t)(λ2 − t)(λ3 − t) (λ1 − t)(λ2 − t) 2<br />
(λ1 − t)(λ2 − t)(λ3 − t) (λ1 − t)(λ2 − t) 2<br />
⎛ ⎞<br />
λ<br />
⎛ ⎞<br />
λ 1<br />
⎝ λ ⎠<br />
⎝ λ ⎠<br />
λ<br />
λ<br />
(λ − t) 3<br />
(λ − t) 3<br />
λ − t (λ − t) 2<br />
λ2<br />
⎝<br />
λ1<br />
λ2<br />
λ2<br />
⎠<br />
(λ1 − t)(λ2 − t) 2<br />
(λ1 − t)(λ2 − t)<br />
⎛ ⎞<br />
λ 1<br />
⎝ λ 1⎠<br />
λ<br />
(λ − t) 3<br />
(λ − t) 3
n = 4 (λ−t) 4 4 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 <br />
<br />
⎛<br />
⎞ ⎛<br />
⎞ ⎛<br />
⎞<br />
λ 1<br />
λ 1<br />
λ<br />
⎟<br />
1⎠<br />
λ<br />
⎜<br />
⎝<br />
(λ − t) 4<br />
⎛<br />
λ 1<br />
⎜<br />
⎝<br />
λ<br />
(λ − t) 2<br />
λ<br />
λ<br />
⎞<br />
⎟<br />
⎠<br />
⎜<br />
⎝<br />
λ 1<br />
(λ − t) 3<br />
⎛<br />
λ<br />
⎜ λ<br />
⎝<br />
λ − t<br />
λ 1<br />
λ<br />
n((α − λι) r ) r <br />
<br />
⎛<br />
2 0 0<br />
⎞<br />
0<br />
⎜<br />
A = ⎜3<br />
⎝0<br />
2<br />
0<br />
0<br />
2<br />
−2 ⎟<br />
0 ⎠<br />
0 0 2 2<br />
χA(t) = (2 − t) 4 <br />
⎛<br />
0 0 0<br />
⎞<br />
0<br />
⎜<br />
A − 2I = ⎜3<br />
⎝0<br />
0<br />
0<br />
0<br />
0<br />
−2 ⎟<br />
0 ⎠<br />
0 0 2 0<br />
λ<br />
λ<br />
λ<br />
⎟<br />
⎠<br />
⎞<br />
⎟<br />
⎠<br />
⎜<br />
⎝<br />
λ 1<br />
λ<br />
(λ − t) 2<br />
⎟<br />
λ 1⎠<br />
λ<br />
(A − 2I) 2 ⎛<br />
0 0 0<br />
⎞<br />
0<br />
⎜<br />
= ⎜0<br />
⎝0<br />
0<br />
0<br />
−4<br />
0<br />
0 ⎟<br />
0⎠<br />
0 0 0 0<br />
A mA(t) = (2 − t) 3 n(A − 2I) = 2 <br />
A <br />
⎛<br />
2 1 0<br />
⎞<br />
0<br />
⎜<br />
JNF = ⎜0<br />
⎝0<br />
2<br />
0<br />
1<br />
2<br />
0 ⎟<br />
0⎠<br />
0 0 0 2<br />
.<br />
v3 ∈ ker(A − 2I) 2 v3 = (0, 0, 1, 0) T A − 2I<br />
⎛ ⎞<br />
0<br />
⎜<br />
v3 = ⎜0<br />
⎟<br />
⎝1⎠<br />
0<br />
↦→ v2<br />
⎛ ⎞<br />
0<br />
⎜<br />
= ⎜0<br />
⎟<br />
⎝0⎠<br />
2<br />
↦→ v1<br />
⎛ ⎞<br />
0<br />
⎜<br />
= ⎜−4<br />
⎟<br />
⎝ 0 ⎠ ↦→ 0.<br />
0<br />
v4 v4 = (2, 0, 0, 3) T
V F V ∗ = L(V, F) <br />
V V ∗ V <br />
V F B = {e1, . . . , en} V <br />
B ∗ = {ε1, . . . , εn} V ∗ B εj(ek) = δjk<br />
n j=1 λjεj = 0 k = 1, . . . , n λk = ( n j=1 λjεj)(ek) = 0 <br />
B∗ ε ∈ V ∗ ε = n j=1 ε(ej)εj B∗ V ∗ ε = n<br />
j=1 ajεj v = n<br />
j=1 xjej <br />
ε(v) =<br />
n<br />
ajxj = ⎛ ⎞<br />
x1<br />
⎜<br />
a1 . . .<br />
<br />
⎟<br />
an ⎝ ⎠ .<br />
j=1<br />
F n n<br />
U ≤ V U ◦ = {ε ∈ V ∗ : ε(u) = 0 ∀u ∈ U} U ◦ U<br />
V ∗ <br />
U ≤ V U ◦ ≤ V ∗ <br />
U ≤ V dim U + dim U ◦ = dim V <br />
<br />
U ≤ V e1, . . . , ek U B = {e1, . . . , ek, . . . , en} <br />
V U ◦ = 〈εk+1, . . . , εn〉 ε1, . . . , εn V ∗ B<br />
i > k εi(ej) = 0 j ≤ k εi ∈ U ◦ ε ∈ U ◦ ε = n<br />
j=1 λjεj <br />
j ≤ k λj = ε(ej) = 0 ε ∈ 〈εk+1, . . . , εn〉<br />
U, V F U α −→ V <br />
V ∗ α∗<br />
−→ U ∗ α∗ (ε) = ε ◦ α ε ∈ V ∗ α<br />
ε ◦ α : U → F α ∗ ∈ U ∗ θ1, θ2 ∈ V ∗ <br />
α ∗ (θ1 + θ2) = (θ1 + θ2) ◦ α<br />
= θ1 ◦ α + θ2 ◦ α<br />
xn
λ ∈ F θ ∈ V ∗ <br />
= α ∗ (θ1) + α ∗ (θ2)<br />
α ∗ (λθ) = (λθ) ◦ α = λ(θ ◦ α) = λα ∗ (θ).<br />
U, V F B, C <br />
B ∗ , C ∗ U ∗ , V ∗ B, C α ∈ L(U, V ) <br />
α ∗ ∈ L(U ∗ , V ∗ ) [α ∗ ]C ∗ ,B ∗ = [α]T B,C <br />
B = {b1, . . . , bn} C = {c1, . . . , cm} B ∗ = {β1, . . . , βn} C ∗ =<br />
{γ1, . . . , γm} A = [α]B,C α(bj) = n<br />
i=1 aijci <br />
α ∗ (γr)(bs) = γr(α(bs))<br />
<br />
n<br />
= γr<br />
=<br />
=<br />
i=1<br />
aisci<br />
n<br />
aisγr(ci)<br />
i=1<br />
n<br />
i=1<br />
= ars<br />
<br />
n<br />
=<br />
i=1<br />
aisδri<br />
ariβi<br />
<br />
<br />
(bs)<br />
s = 1, . . . , n α ∗ (γr) = n<br />
i=1 ariβi [α ∗ ]C ∗ ,B ∗ = AT <br />
det α ∗ = det α χα ∗ = χα mα ∗ = mα det A T = det A<br />
p(A T ) = p(A) T p<br />
U, V F α ∈ L(U, V )<br />
α ∗ ∈ L(V ∗ , U ∗ ) ker α ∗ = (Im α) ◦ α ∗ <br />
α <br />
ε ∈ V ∗ ε ∈ ker α ∗ α ∗ (ε) U ε ◦ α <br />
U ε ∈ Im(α) ◦ α ∗ ker α ∗ = {0} <br />
(Im α) ◦ = {0} Im α = V α <br />
α ∈ L(U, V ) rank α = rank α ∗ A ∈ Mm,n(F) rank A =<br />
rank A T <br />
<br />
rank α ∗ = dim V ∗ − n(α ∗ )<br />
= dim V − dim(Im α) ◦<br />
= dim V − (dim V − dim Im α<br />
= rank α
Im α ∗ = (ker α) ◦ <br />
V ∗ × V → F, (ε, v) ↦→ ε(v) 〈ε|v〉 <br />
U α −→ V V ∗ α∗<br />
−→ U ∗ <br />
〈α ∗ (ε)|u〉 = 〈ε|α(u)〉<br />
u ∈ U ε ∈ V ∗ ˆ : V → V ∗∗ , v ↦→ ˆv ˆv(ε) = ε(v)<br />
V F ˆ : V → V ∗∗ , v ↦→ ˆv <br />
ˆv(ε) = ε(v) <br />
ˆv : V ∗ → F ˆ <br />
<br />
(λ1v1 + λ2v2)(ε) = ε(λ1v1 + λ2v2) = λ1ε(v1) + λ2ε(v2)<br />
= λ1ˆv1(ε) + λ2ˆv2(ε)<br />
= (λ1ˆv1 + λ2ˆv2)(ε)<br />
ε ∈ V ∗ ˆ V <br />
dim V = dim V ∗∗ e1 = 0, e1 ∈ V e1, . . . , en V ε1, . . . , εn <br />
V ∗ ê1(ε1) = ε1(e1) = 1 ê1 = 0<br />
<br />
ε1, . . . , εn V ∗ E1, . . . , En V ∗∗ <br />
Ej = êj ej ∈ V ε1, . . . , εn V ∗ <br />
e1, . . . , en V <br />
V U ≤ V V V ∗∗ <br />
U = U ◦◦ Û = U ◦◦ <br />
U ≤ U ◦◦ u ∈ U ε(u) = 0 ε ∈ U ◦ û(ε) = 0<br />
ε ∈ U ◦ û ∈ U ◦◦ dim U = dim U ◦◦ U = U ◦◦ <br />
U1, U2 ≤ V dim V <br />
(U1 + U2) ◦ = U ◦ 1 ∩ U ◦ 2 <br />
(U1 ∩ U2) ◦ = U ◦ 1 + U ◦ 2 <br />
V V =<br />
P (R) V ∗ = R N <br />
P (R) = 〈p0, p1, . . .〉 ε ∈ V ∗ <br />
(ε(p0), ε(p1), . . . )
U, V F ψ : U × V → F <br />
u ∈ U ψ(u, v) v <br />
v ∈ V ψ(u, v) u U = V <br />
V <br />
F = R V = R n ψ(x, y) = n<br />
i=1 xiyi = x T y<br />
V = F n A ∈ Mn(F) ψ(u, v) = u T Av<br />
V F dim V = n B = {v1, . . . , vn}<br />
V ψ V B A =<br />
(ψ(vi, vj)) = [ψ]B<br />
ψ V B V ψ(u, v) =<br />
[u] T B [ψ]B[v]B u, v ∈ V <br />
B = {v1, . . . , vn} u = n<br />
i=1 aivi v = n<br />
i=1 bivi <br />
ψ(u, v) = ψ( aivi, bjvj)<br />
= <br />
aibjψ(vi, vj)<br />
i,j<br />
⎛<br />
b1<br />
bn<br />
⎞<br />
= ⎜<br />
a1 · · ·<br />
<br />
⎟<br />
an [ψ]B ⎝ ⎠<br />
= [u] T B[ψ]B[v]B.<br />
[ψ]B u, v ∈ V ψ(u, v) =<br />
[u] T B A[v]B u, v ∈ V u = vi v = vj Aij = ψ(vi, vj)<br />
B = {v1, . . . , vn}, B ′ = {v ′ 1 , . . . , v′ n} V P <br />
B B ′ <br />
v ′ j =<br />
n<br />
i=1<br />
pijvi<br />
[ψ]B ′ = P T [ψ]BP <br />
[v]B = P [v]B ′
u, v ∈ V <br />
ψ(u, v) = [u] T B[ψ]B[v]B<br />
[ψ]B ′ = P T [ψ]BP [ψ]B ′<br />
= (P [u]B ′)T [ψ]B(P [v]B ′)<br />
= [u] T B ′P T [ψ]BP [v]B ′,<br />
A, B B = P T AP <br />
P <br />
<br />
Mn(R)<br />
<br />
<br />
ψ ψ(u, v) = ψ(v, u) u, v ∈ V <br />
A = [ψ]B A = A T <br />
P T AP = D P A = A T <br />
V Q : V → R <br />
V Q(λv) = λ 2 Q(v) λ ∈ R, v ∈ V <br />
ψ V Q(v) + Q(w) + 2ψ(v, w) = Q(v + w) v, w ∈ V <br />
ψ Q(v) = ψ(v, v) <br />
(Q(v + w) − Q(v) − Q(w))<br />
Q ψ(v, w) = 1<br />
2<br />
<br />
⎛<br />
Ip<br />
⎝ −Iq<br />
⎞<br />
0<br />
⎠ .<br />
0 0<br />
<br />
dim V = n <br />
ψ(u, v) = 0 u, v ∈ V [ψ]B = 0 <br />
B e ∈ V ψ(e, e) = 0<br />
2ψ(u, v) = ψ(u + v, u + v) − ψ(u, u) − ψ(v, v) 0 u, v ∈ V <br />
W = {v ∈ V : ψ(e, v) = 0} V = 〈e〉 ⊕ W v ∈ V v = λe + (v − λe) <br />
〈e〉 ∩ W = {0}<br />
λ ∈ R λ v − λe ∈ W λ = ψ(e,v)<br />
ψ(e,e)<br />
ψ(e, λe) = 0 λ = 0 ψ ′ ψ W <br />
e2, . . . , en W ψ ′ <br />
ψ ′ <br />
⎛<br />
⎜<br />
⎝<br />
d2<br />
<br />
0 dn<br />
⎞<br />
0<br />
⎟<br />
⎠ .
B0 = {e1, e2, . . . , en} e1 = e [ψ]B0 <br />
⎛<br />
d1<br />
⎜ 0<br />
⎜<br />
⎝<br />
0<br />
d2<br />
⎞<br />
0<br />
⎟<br />
⎠<br />
0 dn<br />
,<br />
d1 = ψ(e, e) <br />
B0 d1, . . . , dp > 0 dp+1, . . . , dp+q < 0 di = 0 <br />
1<br />
i > p + q 1 ≤ i ≤ p + q ei √|di| ei B <br />
rank ψ = p + q<br />
⎛<br />
Ip<br />
[ψ]B = ⎝ −Iq<br />
⎞<br />
0<br />
⎠ .<br />
0 0<br />
ψ s(ψ) = p − q<br />
p = 1<br />
2<br />
<br />
<br />
1<br />
(r + s) q = 2 (r − s) (p, q) <br />
ψ <br />
<br />
⎛<br />
Ip<br />
⎝ −Iq<br />
⎞<br />
0<br />
⎠<br />
⎛<br />
Ip ′<br />
⎝ −Iq<br />
0<br />
0 0<br />
′<br />
⎞<br />
⎠<br />
0 0<br />
p = p ′ q = q ′ <br />
<br />
B = {v1, . . . , vp, vp+1, . . . , vp+q, vp+q+1, . . . , vn} <br />
⎛<br />
Ip<br />
[ψ]B = ⎝ −Iq<br />
⎞<br />
0<br />
⎠ .<br />
0 0<br />
X = 〈v1, . . . , vp〉 Y = 〈vp+1, . . . , vn〉 ψ X <br />
p V ψ <br />
Q( p i=1 λivi) = p i=1 λ2i ≥ 0 p i=1 λivi = 0 ψ <br />
X X ′ ≤ V dim X ′ = p ′ ψ X ′ <br />
X ′ ∩ Y = {0} ψ Y dim X ′ + dim Y ≤ n <br />
p ′ = dim X ′ ≤ n − dim Y = p<br />
N = 〈vp+1, . . . , vp+q〉 ψ N q <br />
V ψ <br />
p q <br />
p ψ X <br />
p ψ q N
t = min{p, q} ψ 〈v1 +<br />
vp+1, . . . , vt + vp+t, vp+q+1, . . . , vn〉 0 n − max{p, q} <br />
ψ 0 ψ = 0 U ≤ V U ∩ X = {0} = U ∩ N<br />
dim U ≤ n − p dim U ≤ n − q<br />
ψ {v ∈ V : ψ(v, w) = 0 ∀w ∈ V } ker ψ =<br />
〈vp+q+1, . . . , vn〉<br />
ψ ker ψ = {0} <br />
[ψ]B B n = p + q<br />
Q V = R 3 <br />
Q(x1, x2, x3) = x 2 1 + x 2 2 + 2x 2 3 + 2x1x2 + 2x1x3 − 2x2x3.<br />
Q <br />
<br />
⎛<br />
1 1<br />
⎞<br />
1<br />
A = ⎝1<br />
1 −1⎠<br />
.<br />
1 −1 2<br />
Q(x1, x2, x3) = x 2 1 + x 2 2 + 2x 2 3 + 2x1x2 + 2x1x3 − 2x2x3<br />
rank(Q) = 3 s(Q) = 2 − 1 = 1 <br />
= (x1 + x2 + x3) 2 + x 2 3 − 4x2x3<br />
= (x1 + x2 + x3) 2 + (x3 − 2x2) 2 − (2x2) 2<br />
P −1 ⎛<br />
1 1<br />
⎞<br />
1<br />
= ⎝0<br />
−2 1⎠<br />
P<br />
0 2 0<br />
T ⎛<br />
1 0<br />
⎞<br />
0<br />
AP = ⎝0<br />
1 0 ⎠<br />
0 0 −1<br />
P P T AP <br />
<br />
<br />
P = E1 · · · Ek<br />
A → E T 1 AE1 → . . . → E T k · · · ET 1 AE1 · · · Ek = D<br />
e1 Q(e1) = 0 e1 = (1, 0, 0) T <br />
Q(e1) = 1 W = {v ∈ V : ψ(e, v) = 0} = {(a, b, c) T : a + b + c = 0} <br />
e T 1 A = (1, 1, 1) e2 ∈ W Q(e2) = 0 e2 = (1, 0, −1) Q(e2) = 1<br />
e3 ∈ W ψ(e2, e3) = 0 e3 = (a, b, c) T a + b + c = 0<br />
2b − c = 0 e T 2 A = (0, 2, −1) e3 = 1<br />
2 (−3, 1, 2)T Q(e3) = −1<br />
s(Q)
C <br />
e1, . . . , en ψ <br />
⎛ ⎞<br />
d1 0<br />
⎜<br />
⎝<br />
⎟<br />
⎠ .<br />
0 dn<br />
d1, . . . , dr = 0 di = 0 i > r <br />
ej 1 √ ej 1 ≤ j ≤ r ψ <br />
dj<br />
<br />
Ir 0<br />
.<br />
0 0<br />
P T AP = <br />
Ir 0<br />
0 0 <br />
P rank A = r<br />
V V <br />
ψ : V × V → C <br />
u ∈ V v ↦→ ψ(u, v) <br />
u, v ∈ V ψ(u, v) =<br />
¯<br />
ψ(v, u)<br />
ψ <br />
ψ(u, v) =<br />
ψ(u, λ1v1 + λ2v2) = λ1ψ(u, v1) + λ2ψ(u, v2)<br />
ψ(λ1u1 + λ2u2, v) = ¯ λ1ψ(u1, v) + ¯ λ2ψ(u2, v)<br />
¯<br />
ψ(v, u)<br />
<br />
<br />
ψ V <br />
Q : V → C Q(v) = ψ(v, v) Q(v) ∈ R v ∈ V <br />
Q(λv) = |v| 2 Q(v) ψ <br />
ψ(u, v) = 1<br />
4 (Q(u + v) − Q(u − v) − iQ(u + iv) + iQ(u − iv))<br />
B = {v1, . . . , vn} V ψ B [ψ]B =<br />
(ψ(vi, vj)) = A ψ(u, v) = [ψ] T<br />
B [ψ]B[v]B A = ĀT <br />
¯ P T AP <br />
P <br />
ψ V <br />
B V ψ <br />
⎛<br />
Ip<br />
⎝ −Iq<br />
⎞<br />
0<br />
⎠<br />
0 0<br />
p q ψ B <br />
v1, . . . , vn <br />
n<br />
Q( ξivi) = |ξ1| 2 + · · · + |ξp| 2 − |ξp+1| 2 − · · · − |ξp+q| 2 .<br />
i=1
ψ V p = n<br />
<br />
ψ(u, v) = −ψ(v, u) <br />
u, v ∈ V ψ(u, u) = 0 u ∈ V A = [ψ]B B V <br />
A T = −A A <br />
A <br />
<br />
A = 1<br />
2 (A + AT ) + 1<br />
2 (A − AT )<br />
ψ V <br />
v1, w1, . . . , vm, wm, v2m+1, . . . , vn ψ <br />
<br />
⎛<br />
0 1<br />
⎞<br />
0<br />
⎜<br />
⎜−1<br />
⎜<br />
⎝<br />
0<br />
0<br />
−1<br />
1<br />
0<br />
0<br />
⎟ .<br />
⎟<br />
⎠<br />
0 0<br />
<br />
dim V = n ψ = 0 <br />
v1, w1 ψ(v1, w1) = 1 ψ(w1, v1) = −1 U = 〈v1, w1〉<br />
W = {v ∈ V : ψ(v1, v) = 0 = ψ(w1, v)} V = U ⊕ W v = (av1 + bw1) +<br />
(v − av1 − bw1) a = ψ(v, w1), b = (ψ(v1, v) v − av1 − bw1 ∈ W <br />
U ∩ W = {0} av1 + bw1 ∈ W ψ(av1 + bw1, av1 − bw1) = a 2 + b 2 <br />
W <br />
ψ n = 2m<br />
B v1, . . . , vm, w1, . . . , wm, v2m+1, . . . , vn <br />
⎛<br />
0<br />
⎝−Im<br />
Im<br />
0<br />
⎞<br />
0<br />
0⎠<br />
.<br />
0 0 0<br />
U × V U, V<br />
F ψ : U × V → F <br />
<br />
U = V ∗ ψ : V ∗ × V → F, (α, v) ↦→ α(v)<br />
F = C ¯ V V V <br />
· λ · v = ¯ λv <br />
¯V × V
ψ : U × V → F <br />
ψL : U → V ∗ , u ↦→ (ψL(u) : v ↦→ ψ(u, v))<br />
ψR : V → U ∗ , v ↦→ (ψR(v) : u ↦→ ψ(u, v))<br />
ψ ker ψL = {0} ker ψR = {0} <br />
ψ dim U = dim V dim U ≤ V ∗ = dim V dim V ≤<br />
dim U ∗ = dim U<br />
dim U = dim V ker ψL = {0} ker ψR = {0}<br />
ψ U × V u1, . . . , un U <br />
ψL(u1), . . . , ψL(un) V ∗ v1, . . . , vn V <br />
ψ(ui, vj) = δij<br />
ψ V W ≤ V W ⊥ =<br />
{v ∈ V : ψ(w, v) = 0 ∀w ∈ W } W ⊥ ≤ V dim W + dim W ⊥ = dim V <br />
u1, . . . , un V u1, . . . , um W <br />
v1, . . . , vn W ⊥ = 〈vm+1, . . . , vn〉<br />
<br />
ψL : V → V ∗ , u ↦→ (ψL(u) : V → F, v ↦→ ψ(u, v)).<br />
ψ W ⊥ = (ψL(W )) ◦ <br />
<br />
v ∈ W ⊥ ⇐⇒ ψ(w, v) = 0 ∀w ∈ W<br />
⇐⇒ ψL(w)(v) = 0 ∀w ∈ W<br />
⇐⇒ v ∈ (ψL(W )) ◦ .<br />
dim W + dim W ⊥ = dim W + dim(ψL(W )) ◦<br />
= dim W + dim V − dim ψL(W )<br />
= dim V.
V V <br />
V 〈v, w〉 <br />
(v, w) ∈ V × V V <br />
<br />
〈, 〉 〈v, v〉 > 0 <br />
v ∈ V \ {0}<br />
v v = 〈v, v〉 |v| > 0 v = 0<br />
v, w ∈ V |〈v, w〉| ≤ vw<br />
v = 0 v = 0 <br />
<br />
• t ∈ R <br />
t = 〈v,w〉<br />
v 2 <br />
• t ∈ C <br />
0 ≤ tv − w 2 = t 2 v 2 − 2t〈v, w〉 + w 2 .<br />
0 ≤ tv − w 2 = t¯tv 2 − ¯t〈v, w〉 − t〈v, w〉 + w 2 .<br />
t = 〈v,w〉<br />
v 2 ¯t = 〈v,w〉<br />
v 2 <br />
v, w = 0 θ <br />
θ ∈ [0, 2π)<br />
cos θ = 〈v,w〉<br />
vw<br />
v, w ∈ V v + w ≤ v + w<br />
<br />
v + w 2 = v 2 + 〈v, w〉 + 〈v, w〉 + w 2<br />
≤ v 2 + 2vw + w 2<br />
= (v + w) 2<br />
d(v, w) = v − w V
R n C n <br />
V = C[0, 1] 〈(, f〉, g) = 1<br />
0 f(t)g(t) dt<br />
{e1, . . . , ek} 〈ei, ej〉 = 0 i = j <br />
ej = 1 j <br />
v = k<br />
j=1 λjej <br />
λj = 〈ej, v〉<br />
v1, . . . , vn V <br />
e1, . . . , en V 〈v1, . . . , vk〉 = 〈e1, . . . , ek〉<br />
1 ≤ k ≤ n<br />
e1 = 1<br />
v1 v1 e1, . . . , ek e ′ k+1 = vk+1 −<br />
k<br />
j=1 λjej λj 〈ej, e ′ k+1 〉 = 0 1 ≤ j ≤ k λj = 〈ej, vk+1〉<br />
e ′ k+1 = 0 v1, . . . , vk, vk+1 ek+1 = 1<br />
e ′ k+1 e′ k+1 <br />
〈ej, ek+1〉 = 0 1 ≤ j ≤ k ek+1 = 1 〈e1, . . . , ek+1〉 = 〈v1, . . . , vk+1〉<br />
<br />
λj = 〈ej,vk+1〉<br />
〈ej,ej〉 <br />
<br />
<br />
e1, . . . , ek <br />
e1, . . . , ek, vk+1, . . . , vn V <br />
e1, . . . , ek, ek+1, . . . , en V <br />
A A = RT <br />
R T <br />
A A = UT U <br />
T <br />
R n v1, . . . , vn <br />
A (1) , . . . , A (n) A e1, . . . , en <br />
R R (j) = ej <br />
R T R = I vk = n<br />
j=1 tjkej vk ∈ 〈e1, . . . , ek〉 <br />
T = (tij) A = RT A (k) = n<br />
j=1 tjkR ( j)<br />
A C n <br />
R U Ū T U = I U <br />
V W ≤ V W ⊥ = {v ∈ V : v ⊥<br />
w ∀w ∈ W } W V <br />
V W ≤ V <br />
V = W ⊕ W ⊥
e1, . . . , ek W e1, . . . , ek, ek+1, . . . , en<br />
V ek+1, . . . , en W ⊥ <br />
v ∈ V v = k j=1 λjej + n j=k+1 λjej V = W + W ⊥ W ∩ W ⊥ = {0}<br />
〈v, v〉 = 0 v = 0 <br />
V W ≤ V <br />
π = πW V W π 2 = π W = Im π W ⊥ = ker π<br />
e1, . . . , ek W <br />
e1, . . . , ek, ek+1, . . . , en V v = n j=1 λjej πw(v) =<br />
k j=1 λjej πw(v) = k j=1 〈ej, v〉ej<br />
ιV = πW + π W ⊥ πW π W ⊥ = 0<br />
W ≤ V v ∈ V πW (v) W v<br />
w0 = πW (v) d(w0, v) ≤ d(w, v) w ∈ W <br />
V α ∈ End(V )<br />
α ∗ V α <br />
α ∗ : V → V 〈αv, w〉 = 〈v, α ∗ w〉<br />
v, w ∈ V B V [α ∗ ]B = [α] T<br />
B <br />
B = {e1, . . . , en} V A = [α]B = (aij) <br />
α ∗ [α ∗ ]B = ĀT C = ĀT 1 ≤ i, j ≤ n<br />
<br />
〈αei, ej〉 = 〈<br />
=<br />
n<br />
k=1<br />
akiek, ej〉<br />
n<br />
aki〈ek, ¯ ej〉<br />
k=1<br />
= ¯<br />
aji<br />
= cij<br />
n<br />
= ckj〈ei, ek〉<br />
k=1<br />
= 〈ei,<br />
n<br />
k=1<br />
= 〈ei, α ∗ ej〉<br />
ckjek〉<br />
〈αv, w〉 = 〈v, α ∗ w〉 v, w ∈ V α ∗ <br />
[α ∗ ]B = ĀT <br />
(α + β) ∗ = α ∗ + β ∗ (λα) ∗ = ¯ λα ∗ <br />
α ∗∗ = α ι ∗ = ι (αβ) ∗ = β ∗ α ∗ <br />
〈α ∗ v, w〉 = 〈v, α ∗∗ w〉<br />
<br />
〈α ∗ v, w〉 = 〈w, α ∗ v〉
= 〈αw, v〉<br />
= 〈v, αw〉<br />
〈v, αw − α ∗∗ w〉 = 0 v ∈ V αw = α ∗∗ w w ∈ V α ∗∗ = α<br />
A n × n <br />
• A A T = A ĀT = A A <br />
• A A T = A −1 ĀT = A −1 A <br />
V R C α ∈ End(V )<br />
• α α = α ∗ 〈αv, w〉 = 〈v, αw〉 <br />
v, w ∈ V <br />
• α α ∗ = α −1 〈αv, αw〉 = 〈v, w〉 α<br />
V <br />
• α αα ∗ = α ∗ α<br />
V α ∈ End(V ) B <br />
V α <br />
[α]B <br />
V α ∈<br />
End(V ) α <br />
V α <br />
<br />
V C λ ∈ C e ∈ V<br />
α(e) = λe e = 1 W = 〈e〉 ⊥ W α w ∈ W <br />
〈α(w), e〉 = 〈w, α ∗ (e)〉 = 〈w, α(e)〉 = 〈w, λe〉 = λ〈w, e〉 = 0<br />
α(w) ∈ W w ∈ W <br />
〈α(w), e〉 = 〈w, α ∗ (e)〉 = 〈w, α −1 (e)〉 = 〈w, λ −1 e〉 = λ −1 〈w, e〉 = 0<br />
α(w) ∈ W V = 〈e〉 ⊕ W α| W <br />
W e2, . . . , en W <br />
α B = {e1, e2, . . . , en} <br />
α<br />
[α]B = [α] T<br />
B [α]B <br />
[α]B =<br />
⎛<br />
⎜<br />
⎝<br />
λ1<br />
<br />
0 λn<br />
⎞<br />
0<br />
⎟<br />
⎠ ,<br />
¯ λj = λj 1 ≤ j ≤ n λj [α−1 ]B = [α] T<br />
B ¯ λj = λ −1<br />
j<br />
1 ≤ j ≤ n |λj| = 1
α ∈ End(V ) v1, v2 α <br />
λ1, λ2 v1 ⊥ v2<br />
〈α(v1), v2〉<br />
¯λ1〈v1, v2〉 = 〈α(v1), v2〉 = v1α(v2) = λ2〈v1, v2〉<br />
λ1 = ¯ λ1 = λ2 〈v1, v2〉 = 0 λ1 = λ2 <br />
α α<br />
<br />
<br />
B V [α]B <br />
[α]B α <br />
<br />
V α ∈ End(V ) <br />
V α<br />
<br />
<br />
V α ∈ End(V ) <br />
α <br />
B V <br />
⎛<br />
1 0<br />
⎞<br />
⎜0<br />
1<br />
⎟<br />
θi ∈ R<br />
⎜<br />
[α]B = ⎜<br />
⎝<br />
−1 0<br />
0 −1<br />
cos θ1 sin θ1<br />
− sin θ1 cos θ1<br />
⎟<br />
⎠<br />
A <br />
R n C n <br />
v1, . . . , vn A <br />
P = (v1, . . . , vn) AP = P D D <br />
P −1 AP = D = P T AP ¯ P T AP = D<br />
ψ <br />
V A B<br />
V ψ <br />
A<br />
A <br />
P P −1 AP = D = P T AP <br />
¯P T AP = D
A n × n V <br />
n B α ∈ End(V ) [α]B = A <br />
ψ [ψ]B = A C <br />
A 〈, 〉 V B <br />
〈ei, ej〉 = δij C 〈, 〉 P <br />
P −1 = P T P −1 AP = P T AP [α]C = [ψ]C<br />
ψ φ <br />
V <br />
ψ V ψ<br />
φ <br />
A, C ψ, φ <br />
ψ P P T AP = I ψ P T CP <br />
Q QT P T CP Q <br />
QT P T AP Q = QT IQ = I <br />
Q T P T CP Q = D =<br />
⎛<br />
⎜<br />
⎝<br />
d1<br />
<br />
0 dn<br />
⎞<br />
0<br />
⎟<br />
⎠ .<br />
d1, . . . , dn D det(C − tA)<br />
det(D − tI) <br />
det(D − tI) = det((P Q) 2 (C − tA)(P Q)) = (det(P Q)) 2 det(C − tA),<br />
det(D − tI) det(C − tA) <br />
<br />
V <br />
α ∈ End(V ) α ∗ ∈ End(V ) 〈α(v), w〉 = 〈v, α ∗ (w)〉<br />
w ∈ V φ(w) : V → F, v ↦→ 〈v, w〉 V <br />
¯V → V ∗ , w ↦→ φ(w) ¯ V V λ · v = ¯ λv <br />
V w ′ ∈ V v ↦→ 〈v, w ′ 〉<br />
w ∈ V v ↦→ 〈α(v), w〉 V w ′ = α ∗ (w)<br />
〈α(v), w〉 = 〈v, α ∗ (w)〉 v, w ∈ V α ∗