Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
102 <strong>Linear</strong> <strong>Algebra</strong>, by Hefferon<br />
means that h( β ⃗ i ) = h i,1<br />
⃗ δ1 + · · · + h i,n<br />
⃗ δn . And, by the definition of the representation of a vector with<br />
respect to a basis, the assumption that<br />
⎛ ⎞<br />
1..<br />
⎜ ⎟<br />
Rep B (⃗v) = ⎝c<br />
⎠<br />
c n<br />
means that ⃗v = c 1<br />
⃗ β1 + · · · + c n<br />
⃗ βn . Substituting gives<br />
h(⃗v) = h(c 1 · ⃗β 1 + · · · + c n · ⃗β n )<br />
= c 1 · h( ⃗ β 1 ) + · · · + c n · ⃗β n<br />
= c 1 · (h 1,1<br />
⃗ δ1 + · · · + h m,1<br />
⃗ δm ) + · · · + c n · (h 1,n<br />
⃗ δ1 + · · · + h m,n<br />
⃗ δm )<br />
= (h 1,1 c 1 + · · · + h 1,n c n ) · ⃗δ 1 + · · · + (h m,1 c 1 + · · · + h m,n c n ) · ⃗δ m<br />
and so h(⃗v) is represented as required.<br />
Three.III.1.29<br />
(a) The picture is this.<br />
The images of the vectors<br />
⎛<br />
from the domain’s basis<br />
⎝ 1 ⎞ ⎛<br />
0⎠ ↦→ ⎝ 1 ⎞ ⎛<br />
0⎠<br />
⎝ 0 ⎞ ⎛<br />
1⎠ ↦→ ⎝ 0<br />
⎞ ⎛<br />
cos θ ⎠ ⎝ 0 ⎞ ⎛<br />
0⎠ ↦→ ⎝ 0<br />
⎞<br />
sin θ⎠<br />
0 0 0 − sin θ 1 cos θ<br />
are represented with respect to the codomain’s basis (again, E 3 ) by themselves, so adjoining the<br />
representations to make the matrix gives this.<br />
⎛<br />
Rep E3 ,E 3<br />
(r θ ) = ⎝ 1 0 0<br />
⎞<br />
0 cos θ sin θ⎠<br />
0 − sin θ cos θ<br />
(b) The picture is similar to the one in the prior answer. The images of the vectors from the domain’s<br />
basis<br />
⎛<br />
⎝ 1 ⎞ ⎛<br />
0⎠ ↦→ ⎝ cos θ<br />
⎞ ⎛<br />
0 ⎠ ⎝ 0 ⎞ ⎛<br />
1⎠ ↦→ ⎝ 0 ⎞ ⎛<br />
1⎠<br />
⎝ 0 ⎞ ⎛<br />
0⎠ ↦→ ⎝ − sin θ<br />
⎞<br />
0 ⎠<br />
0 sin θ 0 0 1 cos θ<br />
are represented with respect to the codomain’s<br />
⎛<br />
basis E 3 by<br />
⎞<br />
themselves, so this is the matrix.<br />
cos θ 0 − sin θ<br />
⎝ 0 1 0 ⎠<br />
sin θ 0 cos θ<br />
(c) To a person standing up, with the vertical z-axis, a rotation of the xy-plane that is clockwise<br />
proceeds from the positive y-axis to the positive x-axis. That is, it rotates opposite to the direction<br />
in Example 1.8. The<br />
⎛<br />
images of the vectors from the domain’s basis<br />
⎝ 1 ⎞ ⎛<br />
0⎠ ↦→ ⎝ cos θ<br />
⎞ ⎛<br />
− sin θ⎠<br />
⎝ 0 ⎞ ⎛<br />
1⎠ ↦→ ⎝ sin θ<br />
⎞ ⎛<br />
cos θ⎠<br />
⎝ 0 ⎞ ⎛<br />
0⎠ ↦→ ⎝ 0 ⎞<br />
0⎠<br />
0 0<br />
0 0 1 1<br />
are represented with respect to E 3 by themselves,<br />
⎛<br />
so the matrix<br />
⎞<br />
is this.<br />
cos θ sin θ 0<br />
⎝− sin θ cos θ 0⎠<br />
0 0 1<br />
⎛<br />
⎞<br />
cos θ sin θ 0 0<br />
(d) ⎜− sin θ cos θ 0 0<br />
⎟<br />
⎝ 0 0 1 0⎠<br />
0 0 0 1<br />
Three.III.1.30 (a) Write B U as 〈 β ⃗ 1 , . . . , β ⃗ k 〉 and then B V as 〈 β ⃗ 1 , . . . , β ⃗ k , β ⃗ k+1 , . . . , β ⃗ n 〉. If<br />
⎛ ⎞<br />
1..<br />
⎜ ⎟<br />
Rep BU<br />
(⃗v) = ⎝c<br />
⎠ so that ⃗v = c 1 · ⃗β 1 + · · · + c k · ⃗β k<br />
c k
Change language