Linear Algebra, 2020a

Section III. Basis and Dimension 133
Proof First we will show that a subset with n vectors is linearly independent if
and only if it is a basis. The ‘if’ is trivially true — bases are linearly independent.
‘Only if’ holds because a linearly independent set can be expanded to a basis,
but a basis has n elements, so this expansion is actually the set that we began
with.
To finish, we will show that any subset with n vectors spans the space if and
only if it is a basis. Again, ‘if’ is trivial. ‘Only if’ holds because any spanning
set can be shrunk to a basis, but a basis has n elements and so this shrunken
set is just the one we started with.
QED
The main result of this subsection, that all of the bases in a finite-dimensional
vector space have the same number of elements, is the single most important
result in this book. As Example 2.11 shows, it describes what vector spaces and
subspaces there can be.
One immediate consequence brings us back to when we considered the two
things that could be meant by the term ‘minimal spanning set’. At that point we
defined ‘minimal’ as linearly independent but we noted that another reasonable
interpretation of the term is that a spanning set is ‘minimal’ when it has the
fewest number of elements of any set with the same span. Now that we have
shown that all bases have the same number of elements, we know that the two
senses of ‘minimal’ are equivalent.
Exercises
Assume that all spaces are finite-dimensional unless otherwise stated.
̌ 2.15 Find a basis for, and the dimension of, P 2 .
2.16 Find a basis for, and the dimension of, the solution set of this system.
x 1 − 4x 2 + 3x 3 − x 4 = 0
2x 1 − 8x 2 + 6x 3 − 2x 4 = 0
̌ 2.17 Find a basis for, and the dimension of, each space.
⎛ ⎞
x
(a) { ⎜y
⎟
⎝ z ⎠ ∈ R4 | x − w + z = 0}
w
(b) the set of 5×5 matrices whose only nonzero entries are on the diagonal (e.g.,
in entry 1, 1 and 2, 2, etc.)
(c) {a 0 + a 1 x + a 2 x 2 + a 3 x 3 | a 0 + a 1 = 0 and a 2 − 2a 3 = 0} ⊆ P 3
2.18 Find a basis for, and the dimension of, M 2×2 , the vector space of 2×2 matrices.
2.19 Find the dimension of the vector space of matrices
( ) a b
subject to each condition.
(a) a, b, c, d ∈ R
(b) a − b + 2c = 0 and d ∈ R
c
d