Linear Algebra

Section III. Computing Linear Maps 203
(c) � β1 ↦→ � β2, � β2 ↦→ � β3, � β3 ↦→ �0, � β4 ↦→ �0
1.21 Example 1.8 shows how to represent the rotation transformation of the plane
with respect to the standard basis. Express these other transformations also with
respect to the standard basis.
(a) the dilation map ds, which multiplies all vectors by the same scalar s
(b) the reflection map fℓ, which reflects all all vectors across a line ℓ through the
origin
� 1.22 Consider a linear transformation of R 2 determined by these two.
� � � � � � � �
1 2 1 −1
↦→
↦→
1 0 0 0
(a) Represent this transformation with respect to the standard bases.
(b) Where does the transformation send this vector?
� �
0
5
(c) Represent this transformation with respect to these bases.
� � � � � � � �
1 1
2 −1
B = 〈 , 〉 D = 〈 , 〉
−1 1
2 1
(d) Using B from the prior item, represent the transformation with respect to
B,B.
1.23 Suppose that h: V → W is nonsingular so that by Theorem 2.20, for any
basis B = 〈 � β1,..., � βn〉 ⊂V the image h(B) =〈h( � β1),...,h( � βn)〉 is a basis for
W .
(a) Represent the map h with respect to B,h(B).
(b) For a member �v of the domain, where the representation of �v has components
c1, ... , cn, represent the image vector h(�v) with respect to the image basis h(B).
1.24 Give a formula for the product of a matrix and �ei, the column vector that is
all zeroes except for a single one in the i-th position.
� 1.25 For each vector space of functions of one real variable, represent the derivative
transformation with respect to B,B.
(a) {a cos x + b sin x � � a, b ∈ R}, B = 〈cos x, sin x〉
(b) {ae x + be 2x � � a, b ∈ R}, B = 〈e x ,e 2x 〉
(c) {a + bx + ce x + dxe 2x � � a, b, c, d ∈ R}, B = 〈1,x,e x ,xe x 〉
1.26 Find the range of the linear transformation of R 2 represented with respect to
the standard � �bases
by each � matrix. �
� �
1 0
0 0
a b
(a)
(b)
(c) a matrix of the form
0 0
3 2
2a 2b
� 1.27 Can one matrix represent two different linear maps? That is, can RepB,D(h) =
RepB, ˆ D ˆ ( ˆ h)?
1.28 Prove Theorem 1.4.
� 1.29 Example 1.8 shows how to represent rotation of all vectors in the plane through
an angle θ about the origin, with respect to the standard bases.
(a) Rotation of all vectors in three-space through an angle θ about the x-axis is a
transformation of R 3 . Represent it with respect to the standard bases. Arrange
the rotation so that to someone whose feet are at the origin and whose head is
at (1, 0, 0), the movement appears clockwise.