Linear Algebra, 2020a

230 Chapter Three. Maps Between Spaces
⎛
0 1
⎞
3
(b) ⎝ 2 3 4⎠
−2 −1 2
⎛ ⎞
1 1
(c) ⎝2 1⎠
3 1
2.23 Use the method from the prior exercise on this matrix.
⎛
1 0
⎞
−1
⎝2 1 0 ⎠
2 2 2
2.24 Verify that the map represented by this matrix is an isomorphism.
⎛
2 1
⎞
0
⎝3 1 1⎠
7 2 1
2.25 This is an alternative proof of Lemma 2.9. Given an n×n matrix H, fixa
domain V and codomain W of appropriate dimension n, and bases B, D for those
spaces, and consider the map h represented by the matrix.
(a) Show that h is onto if and only if there is at least one Rep B (⃗v) associated by
H with each Rep D (⃗w).
(b) Show that h is one-to-one if and only if there is at most one Rep B (⃗v) associated
by H with each Rep D (⃗w).
(c) Consider the linear system H·Rep B (⃗v) =Rep D (⃗w). Show that H is nonsingular
if and only if there is exactly one solution Rep B (⃗v) for each Rep D (⃗w).
̌ 2.26 Because the rank of a matrix equals the rank of any map it represents, if
one matrix represents two different maps H = Rep B,D (h) =RepˆB, ˆD
(ĥ) (where
h, ĥ: V → W) then the dimension of the range space of h equals the dimension of
the range space of ĥ. Must these equal-dimensional range spaces actually be the
same?
2.27 Let V be an n-dimensional space with bases B and D. Consider a map that
sends, for ⃗v ∈ V, the column vector representing ⃗v with respect to B to the column
vector representing ⃗v with respect to D. Show that map is a linear transformation
of R n .
2.28 Example 2.3 shows that changing the pair of bases can change the map that
a matrix represents, even though the domain and codomain remain the same.
Could the map ever not change? Is there a matrix H, vector spaces V and W,
and associated pairs of bases B 1 ,D 1 and B 2 ,D 2 (with B 1 ≠ B 2 or D 1 ≠ D 2 or
both) such that the map represented by H with respect to B 1 ,D 1 equals the map
represented by H with respect to B 2 ,D 2 ?
̌ 2.29 A square matrix is a diagonal matrix if it is all zeroes except possibly for the
entries on its upper-left to lower-right diagonal — its 1, 1 entry, its 2, 2 entry, etc.
Show that a linear map is an isomorphism if there are bases such that, with respect
to those bases, the map is represented by a diagonal matrix with no zeroes on the
diagonal.