Page 253 - Matrix Analysis & Applied Linear Algebra

248 Chapter 4 Vector Spaces

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2
4.7.7. Let T : → be the linear transformation defined by
T(x, y)=(x +3y, 0, 2x − 4y).

(a) Determine [T] SS , where S and S are the standard bases for
3
2
and , respectively.
3
(b) Determine [T] SS , where S is the basis for obtained by
permuting the standard basis according to S = {e 3 , e 2 , e 1 }.

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4.7.8. Let T be the operator on defined by T(x, y, z)=(x−y, y−x, x−z)
and consider the vector
     
 
1  1 0 1 
1
1
1
,
0
,
v =   and the basis B =       .
2  1 1 0 
(a) Determine [T] B and [v] B .
(b) Compute [T(v)] B , and then verify that [T] B [v] B =[T(v)] B .
n×n n×1
4.7.9. For A ∈ , let T be the linear operator on defined by
T(x)= Ax. That is, T is the operator defined by matrix multiplica-
tion. With respect to the standard basis S, show that [T] S = A.
4.7.10. If T is a linear operator on a space V with basis B, explain why
k
[T ] B =[T] k for all nonnegative integers k.
B
2
4.7.11. Let P be the projector that maps each point v ∈ to its orthogonal
projection on the line y = x as depicted in Figure 4.7.4.
y = x
v

P(v)

Figure 4.7.4
(a) Determine the coordinate matrix of P with respect to the stan-
dard basis.
α

(b) Determine the orthogonal projection of v = onto the line
β
y = x.