Linear Algebra, 2020a

Section I. Isomorphisms 189
Associate ⃗β 1 with ⃗δ 1 , etc. Extending that gives another isomorphism.
⎛ ⎞
( )
a
a b
f
= a⃗β 1 + b⃗β 2 + c⃗β 3 + d⃗β
3
b
4 ↦−→ a ⃗δ 1 + b⃗δ 2 + c⃗δ 3 + d⃗δ 4 = ⎜ ⎟
c d
⎝d⎠
c
We close with a recap. Recall that the first chapter defines two matrices to be
row equivalent if they can be derived from each other by row operations. There
we showed that relation is an equivalence and so the collection of matrices is
partitioned into classes, where all the matrices that are row equivalent together
fall into a single class. Then for insight into which matrices are in each class we
gave representatives for the classes, the reduced echelon form matrices.
In this section we have followed that pattern except that the notion here
of “the same” is vector space isomorphism. We defined it and established some
properties, including that it is an equivalence. Then, as before, we developed
a list of class representatives to help us understand the partition — it classifies
vector spaces by dimension.
In Chapter Two, with the definition of vector spaces, we seemed to have
opened up our studies to many examples of new structures besides the familiar
R n ’s. We now know that isn’t the case. Any finite-dimensional vector space is
actually “the same” as a real space.
Exercises
̌ 2.10 Decide if the spaces are isomorphic.
(a) R 2 , R 4 (b) P 5 , R 5 (c) M 2×3 , R 6 (d) P 5 , M 2×3
(e) M 2×k , M k×2
2.11 Which of these spaces are isomorphic to each other?
(a) R 3 (b) M 2×2 (c) P 3 (d) R 4 (e) P 2
̌ 2.12 Consider the isomorphism Rep B (·): P 1 → R 2 where B = 〈1, 1 + x〉. Find the
image of each of these elements of the domain.
(a) 3 − 2x; (b) 2 + 2x; (c) x
2.13 For which n is the space isomorphic to R n ?
(a) P 4
(b) P 1
(c) M 2×3
(d) the plane 2x − y + z = 0 subset of R 3
(e) the vector space of linear combinations of three letters {ax + by + cz | a, b, c ∈ R}
̌ 2.14 Show that if m ≠ n then R m ̸∼ = R n .
̌ 2.15 Is M m×n = ∼ Mn×m ?
̌ 2.16 Are any two planes through the origin in R 3 isomorphic?
2.17 Find a set of equivalence class representatives other than the set of R n ’s.