Linear Algebra Exercises-n-Answers.pdf

108 Linear Algebra, by Hefferon
(e) Apply that (r + s) · g(⃗v) = r · g(⃗v) + s · g(⃗v).
(f) Apply the prior two items with r = 1 and s = −1.
(g) Apply that r · (g(⃗v) + h(⃗v)) = r · g(⃗v) + r · h(⃗v).
(h) Apply that (rs) · g(⃗v) = r · (s · g(⃗v)).
Three.IV.1.10 For any V, W with bases B, D, the (appropriately-sized) zero matrix represents this
map.
⃗β 1 ↦→ 0 · ⃗δ 1 + · · · + 0 · ⃗δ m · · · βn ⃗ ↦→ 0 · ⃗δ 1 + · · · + 0 · ⃗δ m
This is the zero map.
There are no other matrices that represent only one map. For, suppose that H is not the zero
matrix. Then it has a nonzero entry; assume that h i,j ≠ 0. With respect to bases B, D, it represents
h 1 : V → W sending
⃗β j ↦→ h 1,j
⃗ δ1 + · · · + h i,j
⃗ δi + · · · + h m,j
⃗ δm
and with respcet to B, 2 · D it also represents h 2 : V → W sending
⃗β j ↦→ h 1,j · (2 ⃗ δ 1 ) + · · · + h i,j · (2 ⃗ δ i ) + · · · + h m,j · (2 ⃗ δ m )
(the notation 2·D means to double all of the members of D). These maps are easily seen to be unequal.
Three.IV.1.11 Fix bases B and D for V and W , and consider Rep B,D : L(V, W ) → M m×n associating
each linear map with the matrix representing that map h ↦→ Rep B,D (h). From the prior section we
know that (under fixed bases) the matrices correspond to linear maps, so the representation map is
one-to-one and onto. That it preserves linear operations is Theorem 1.5.
Three.IV.1.12 Fix bases and represent the transformations with 2×2 matrices. The space of matrices
M 2×2 has dimension four, and hence the above six-element set is linearly dependent. By the prior
exercise that extends to a dependence of maps. (The misleading part is only that there are six
transformations, not five, so that we have more than we need to give the existence of the dependence.)
Three.IV.1.13 That the trace of a sum is the sum of the traces holds because both trace(H + G) and
trace(H) + trace(G) are the sum of h 1,1 + g 1,1 with h 2,2 + g 2,2 , etc. For scalar multiplication we have
trace(r · H) = r · trace(H); the proof is easy. Thus the trace map is a homomorphism from M n×n to
R.
Three.IV.1.14 (a) The i, j entry of (G + H) trans is g j,i + h j,i . That is also the i, j entry of G trans +
H trans .
(b) The i, j entry of (r · H) trans is rh j,i , which is also the i, j entry of r · H trans .
Three.IV.1.15 (a) For H + H trans , the i, j entry is h i,j + h j,i and the j, i entry of is h j,i + h i,j . The
two are equal and thus H + H trans is symmetric.
Every symmetric matrix does have that form, since it can be written H = (1/2) · (H + H trans ).
(b) The set of symmetric matrices is nonempty as it contains the zero matrix. Clearly a scalar
multiple of a symmetric matrix is symmetric. A sum H + G of two symmetric matrices is symmetric
because h i,j + g i,j = h j,i + g j,i (since h i,j = h j,i and g i,j = g j,i ). Thus the subset is nonempty and
closed under the inherited operations, and so it is a subspace.
Three.IV.1.16 (a) Scalar multiplication leaves the rank of a matrix unchanged except that multiplication
by zero leaves the matrix with rank zero. (This follows from the first theorem of the book,
that multiplying a row by a nonzero scalar doesn’t change the solution set of the associated linear
system.)
(b) A sum of rank n matrices can have rank less than n. For instance, for any matrix H, the sum
H + (−1) · H has rank zero.
A sum of rank n matrices can have rank greater than n. Here are rank one matrices that sum
to a rank two matrix. ( ) ( ) ( )
1 0 0 0 1 0
+ =
0 0 0 1 0 1
Subsection Three.IV.2: Matrix Multiplication
Three.IV.2.14
(a)
( )
0 15.5
0 −19
(b)
(
2 −1
)
−1
17 −1 −1
(c) Not defined.
(d)
( )
1 0
0 1