Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 123
( )
⃗ cos(2θ)
δ1 =
sin(2θ)
⃗e 2
↦−→
( )
⃗ sin(2θ)
δ2 =
− cos(2θ)
f
⃗e 1
This map reflects vectors over that line. Since reflections are self-inverse, the answer to the question
is: the original map reflects about the line through the origin with angle of elevation θ. (Of course, it
does this to any basis.)
Three.V.1.14
Three.V.1.15
The appropriately-sized identity matrix.
Each is true if and only if the matrix is nonsingular.
Three.V.1.16 What remains to be shown is that left multiplication by a reduction matrix represents
a change from another basis to B = 〈 β ⃗ 1 , . . . , β ⃗ n 〉.
Application of a row-multiplication matrix M i (k) translates a representation with respect to the
basis 〈 β ⃗ 1 , . . . , kβ ⃗ i , . . . , β ⃗ n 〉 to one with respect to B, as here.
⃗v = c 1 · ⃗β 1 + · · · + c i · (kβ ⃗ i ) + · · · + c n · ⃗β n ↦→ c 1 · ⃗β 1 + · · · + (kc i ) · ⃗β i + · · · + c n · ⃗β n = ⃗v
Applying a row-swap matrix P i,j translates a representation with respect to 〈 β ⃗ 1 , . . . , β ⃗ j , . . . , β ⃗ i , . . . , β ⃗ n 〉
to one with respect to 〈 β ⃗ 1 , . . . , β ⃗ i , . . . , β ⃗ j , . . . , β ⃗ n 〉. Finally, applying a row-combination matrix C i,j (k)
changes a representation with respect to 〈 β ⃗ 1 , . . . , β ⃗ i + kβ ⃗ j , . . . , β ⃗ j , . . . , β ⃗ n 〉 to one with respect to B.
⃗v = c 1 · ⃗β 1 + · · · + c i · ( ⃗ β i + k ⃗ β j ) + · · · + c j
⃗ βj + · · · + c n · ⃗β n
↦→ c 1 · ⃗β 1 + · · · + c i · ⃗β i + · · · + (kc i + c j ) · ⃗β j + · · · + c n · ⃗β n = ⃗v
(As in the part of the proof in the body of this subsection, the various conditions on the row operations,
e.g., that the scalar k is nonzero, assure that these are all bases.)
Three.V.1.17 Taking H as a change of basis matrix H = Rep B,En (id), its columns are
⎛ ⎞
h 1,i
⎜ . ⎟
⎝ . ⎠ = Rep En
(id( β ⃗ i )) = Rep En
( β ⃗ i )
h n,i
and, because representations with respect to the standard basis are transparent, we have this.
⎛ ⎞
h 1,i
⎜
⎝
⎟
. ⎠ = β ⃗ i
h n,i
That is, the basis is the one composed of the columns of H.
Three.V.1.18 (a) We can change the starting vector representation to the ending one through a
sequence of row operations. The proof tells us what how the bases change. We start by swapping
the first and second rows of the representation with respect to B to get a representation with resepect
to a new basis B 1 .
⎛ ⎞
1
Rep B1
(1 − x + 3x 2 − x 3 ) = ⎜0
⎟
⎝ B
1
1 = 〈1 − x, 1 + x, x 2 + x 3 , x 2 − x 3 〉
2
⎠B 1
We next add −2 times the third row of
⎛
the
⎞
vector representation to the fourth row.
1
Rep B3
(1 − x + 3x 2 − x 3 ) = ⎜0
⎟
⎝ B
1
2 = 〈1 − x, 1 + x, 3x 2 − x 3 , x 2 − x 3 〉
0
⎠B 2
(The third element of B 2 is the third element of B 1 minus −2 times the fourth element of B 1 .) Now
we can finish by doubling the third row.
⎛ ⎞
1
Rep D (1 − x + 3x 2 − x 3 ) = ⎜0
⎟
⎝2⎠
D = 〈1 − x, 1 + x, (3x 2 − x 3 )/2, x 2 − x 3 〉
0
D