Page 38 - Linear Algebra Done Right

Chapter 2. Finite-Dimensional Vector Spaces
24

(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0)
4
is linearly independent in F , as you should verify. The reasoning in the
previous paragraph shows that (v 1 ,...,v m ) is linearly independent if
and only if each vector in span(v 1 ,...,v m ) has only one representation
as a linear combination of (v 1 ,...,v m ).
Most linear algebra For another example of a linearly independent list, fix a nonnegative
m
texts define linearly integer m. Then (1,z,...,z ) is linearly independent in P(F). To verify
independent sets this, suppose that a 0 ,a 1 ,...,a m ∈ F are such that
instead of linearly
2.3 a 0 + a 1 z + ··· + a m z m = 0
independent lists. With
that definition, the set
for every z ∈ F. If at least one of the coefficients a 0 ,a 1 ,...,a m were
{(0, 1), (0, 1), (1, 0)} is
nonzero, then 2.3 could be satisfied by at most m distinct values of z (if
linearly independent in you are unfamiliar with this fact, just believe it for now; we will prove
2
F because it equals the
it in Chapter 4); this contradiction shows that all the coefficients in 2.3
set {(0, 1), (1, 0)}. With m
equal 0. Hence (1,z,...,z ) is linearly independent, as claimed.
our definition, the list
A list of vectors in V is called linearly dependent if it is not lin-
(0, 1), (0, 1), (1, 0) is early independent. In other words, a list (v 1 ,...,v m ) of vectors in V
not linearly is linearly dependent if there exist a 1 ,...,a m ∈ F, not all 0, such that
independent (because 1 a 1 v 1 +· · ·+ a m v m = 0. For example, (2, 3, 1), (1, −1, 2), (7, 3, 8) is

times the first vector
3
linearly dependent in F because
plus −1 times the
second vector plus 0 2(2, 3, 1) + 3(1, −1, 2) + (−1)(7, 3, 8) = (0, 0, 0).
times the third vector
equals 0). By dealing As another example, any list of vectors containing the 0 vector is lin-
with lists instead of early dependent (why?).
sets, we will avoid You should verify that a list (v) of length 1 is linearly independent if
some problems and only if v = 0. You should also verify that a list of length 2 is linearly
associated with the independent if and only if neither vector is a scalar multiple of the
usual approach. other. Caution: a list of length three or more may be linearly dependent
even though no vector in the list is a scalar multiple of any other vector
in the list, as shown by the example in the previous paragraph.
If some vectors are removed from a linearly independent list, the
remaining list is also linearly independent, as you should verify. To
allow this to remain true even if we remove all the vectors, we declare
the empty list () to be linearly independent.
The lemma below will often be useful. It states that given a linearly
dependent list of vectors, with the first vector not zero, one of the
vectors is in the span of the previous ones and furthermore we can
throw out that vector without changing the span of the original list.