Math 304 Answers to Selected Problems
1
Section 4.2
5. Find the standard matrix representations for each of the following linear
operators.
(a) L is the linear operator that rotates each x in R2 by 45◦ in the
clockwise direction.
(b) L is the linear operator that reflects each x in R2 about the x1
axis and then rotates it 90◦ in the counterclockwise direction.
(c) L doubles the length of x and then rotates it 30◦ in the counterclockwise direction.
(d) L reflects each vector x about the line x2 = x1 and then projects
it onto the x1 axis.
Answer: For each of these, we will apply L to the standard basis vectors. The resulting vectors are the columns of the matrix representing
the linear operator with respect to the standard basis.
(a)
√ 1/√2
L
=
−1/ 2
√ 0
1/√2
L
=
1
1/ 2
Thus, the matrix is
1
0
√
√ 1/√2 −1/√2
.
1/ 2
1/ 2
(b)
L
L
1
0
0
1
1
0
1
1
0
=
=
Thus, the matrix is
0 1
1 0
.
(c)
√ 3
L
=
1
−1
0
√
L
=
1
3
√
Thus, the matrix is
1
0
3 √
−1
3
1
.
(d)
L
L
Thus, the matrix is
1
0
0
1
0 1
0 0
0
0
1
0
=
=
.
14. The linear transformation L is defined by
L (p(x)) = p0 (x) + p(0)
maps P3 into P2 . Find the matrix representation of L with respect
to the ordered bases [x2 , x, 1] and [2, 1 − x]. For each of the following
vectors in p(x) in P3 , find the coordinates of L (p(x)) with respect to
the ordered basis [2, 1 − x].
(a) x2 + 2x − 3
(d) 4x2 + 2x
Answer: First, we find the matrix representing L with respect to the
two given bases. To do this, we apply L to each of the polynomials in
the first basis:
2
L(x2 ) = 2x
L(x) = 1
L(1) = 1
Next, we find the coordinates for each of the above polynomials in the
second basis. In this case, we can find the coordinates by inspection.
2x = 1(2) − 2(1 − x)
1
1 =
(2) + 0(1 − x)
2
Thus, in basis [2, 1 − x], the polynomial 2x is
.5
nomial 1 is
.
0
1
−2
, and the poly-
Thus, the matrix representation of L with respect to the given matrices
is
1 .5 .5
−2 0 0
Now, we can use this matrix in parts (a) and (d). We just find the
coordinates of the given polynomials with respect to the basis [x2 , x, 1],
and then we multiply by the above matrix.


1
1 .5 .5 
.5
2 =
(a)
−2 0 0
−2
−3
(d)
1 .5 .5
−2 0 0


4
5
 2 =
−8
0
3
15. Let S be the subspace of C[a, b] spanned by ex , xex , and x2 ex . Let D
be the differentiation operator of S. Find the matrix representing D
with respect to [ex , xex , x2 ex ].
Answer: To find the matrix representing D with respect to this basis,
we start by applying D to each basis vector:
D(ex ) = ex
D(xex ) = xex + ex
D(x2 ex ) = x2 ex + 2xex


1
In the given basis, ex corresponds to the vector  0 , xex + ex corre0
 
1

1 , and x2 ex + 2xex corresponds to the vector
sponds to the vector
0
 
0
 2 . Thus, the matrix representing D is
1


1 1 0
 0 1 2 
0 0 1
18. Let E = [u1 , u2 , u3 ] and F = [b1 , b2 ], where
u1 = (1, 0, −1)T ,
u2 = (1, 2, 1)T ,
u3 = (−1, 1, 1)T
and
b1 = (1, −1)T ,
b2 = (2, −1)T
For each of the following linear transformations L from R3 into R2 , find
the matrix representing L with respect to the ordered bases E and F .
(a) L(x) = (x3 , x1 )T
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(c) L(x) = (2x2 , −x1 )T
Answer:
(a) First, we apply L to each of the vectors in the basis E:
−1
L(u1 ) =
1
1
L(u2 ) =
1
1
L(u3 ) =
−1
Next, we change each of these vectors into the basis F . To do this,
we find the transition matrix that converts from the standard
basis to F . The transition
matrix that converts from F to the
1
2
standard basis is S =
. The transition matrix from
−1 −1
−1 −2
−1
the standard basis to F is the S =
. Thus,
1
1
−1
−1 −2
−1
−1
=
=
1
1
1
1
0
F
1
−1 −2
1
−3
=
=
1
1
1
1
2
F
1
−1 −2
1
1
=
=
−1
1
1
−1
0
F
Thus, the matrix representing L with respect to the bases E and
F is
−1 −3 1
0
2 0
(c) First, we apply L to each of the vectors in the basis E:
5
L(u1 ) =
0
−1
4
L(u2 ) =
−1
2
L(u3 ) =
1
Next, we change each of these vectors into the basis F by multiplying by the transition matrix found in part (a):
0
−1
=
F
−1 −2
1
1
0
−1
=
2
−1
4
−1 −2
4
−2
=
=
−1
1
1
−1
3
F
2
−1 −2
2
−4
=
=
1
1
1
1
3
F
Thus, the matrix representing L with respect to the bases E and
F is
2 −2 −4
−1
3
3
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