Linear Algebra, 2020a

190 Chapter Three. Maps Between Spaces
2.18 True or false: between any n-dimensional space and R n there is exactly one
isomorphism.
2.19 Can a vector space be isomorphic to one of its proper subspaces?
̌ 2.20 This subsection shows that for any isomorphism, the inverse map is also an
isomorphism. This subsection also shows that for a fixed basis B of an n-dimensional
vector space V, the map Rep B : V → R n is an isomorphism. Find the inverse of
this map.
̌ 2.21 Prove these facts about matrices.
(a) The row space of a matrix is isomorphic to the column space of its transpose.
(b) The row space of a matrix is isomorphic to its column space.
2.22 Show that the function from Theorem 2.3 is well-defined.
2.23 Is the proof of Theorem 2.3 valid when n = 0?
2.24 For each, decide if it is a set of isomorphism class representatives.
(a) {C k | k ∈ N}
(b) {P k | k ∈ {−1, 0, 1, . . .}}
(c) {M m×n | m, n ∈ N}
2.25 Let f be a correspondence between vector spaces V and W (that is, a map that
is one-to-one and onto). Show that the spaces V and W are isomorphic via f if and
only if there are bases B ⊂ V and D ⊂ W such that corresponding vectors have the
same coordinates: Rep B (⃗v) =Rep D (f(⃗v)).
2.26 Consider the isomorphism Rep B : P 3 → R 4 .
(a) Vectors in a real space are orthogonal if and only if their dot product is zero.
Give a definition of orthogonality for polynomials.
(b) The derivative of a member of P 3 is in P 3 . Give a definition of the derivative
of a vector in R 4 .
̌ 2.27 Does every correspondence between bases, when extended to the spaces, give an
isomorphism? That is, suppose that V is a vector space with basis B = 〈⃗β 1 ,...,⃗β n 〉
and that f: B → W is a correspondence such that D = 〈f(⃗β 1 ),...,f(⃗β n )〉 is basis
for W. Must ˆf: V → W sending ⃗v = c 1
⃗β 1 +···+c n
⃗β n to ˆf(⃗v) =c 1 f(⃗β 1 )+···+c n f(⃗β n )
be an isomorphism?
2.28 (Requires the subsection on Combining Subspaces, which is optional.) Suppose
that V = V 1 ⊕ V 2 and that V is isomorphic to the space U under the map f.
Show that U = f(V 1 ) ⊕ f(V 2 ).
2.29 Show that this is not a well-defined function from the rational numbers to the
integers: with each fraction, associate the value of its numerator.