Linear Algebra, 2020a

Section III. Computing Linear Maps 231
2.30 Describe geometrically the action on R 2 of the map represented with respect
to the standard bases E 2 , E 2 by this matrix.
( ) 3 0
0 2
Do the same for these.
( 1
) 0
( 0
) 1
( 1
) 3
0 0 1 0 0 1
2.31 The fact that for any linear map the rank plus the nullity equals the dimension
of the domain shows that a necessary condition for the existence of a homomorphism
between two spaces, onto the second space, is that there be no gain in dimension.
That is, where h: V → W is onto, the dimension of W must be less than or equal
to the dimension of V.
(a) Show that this (strong) converse holds: no gain in dimension implies that
there is a homomorphism and, further, any matrix with the correct size and
correct rank represents such a map.
(b) Are there bases for R 3 such that this matrix
⎛ ⎞
1 0 0
H = ⎝2 0 0⎠
0 1 0
represents a map from R 3 to R 3 whose range is the xy plane subspace of R 3 ?
2.32 Let V be an n-dimensional space and suppose that ⃗x ∈ R n . Fix a basis
B for V and consider the map h ⃗x : V → R given ⃗v ↦→ ⃗x • Rep B (⃗v) by the dot
product.
(a) Show that this map is linear.
(b) Show that for any linear map g: V → R there is an ⃗x ∈ R n such that g = h ⃗x .
(c) In the prior item we fixed the basis and varied the ⃗x to get all possible linear
maps. Can we get all possible linear maps by fixing an ⃗x and varying the basis?
2.33 Let V, W, X be vector spaces with bases B, C, D.
(a) Suppose that h: V → W is represented with respect to B, C by the matrix H.
Give the matrix representing the scalar multiple rh (where r ∈ R) with respect
to B, C by expressing it in terms of H.
(b) Suppose that h, g: V → W are represented with respect to B, C by H and G.
Give the matrix representing h + g with respect to B, C by expressing it in terms
of H and G.
(c) Suppose that h: V → W is represented with respect to B, C by H and g: W → X
is represented with respect to C, D by G. Give the matrix representing g ◦ h
with respect to B, D by expressing it in terms of H and G.