Page 37 - Linear Algebra Done Right

Span and Linear Independence
for all z ∈ F. The polynomial that is identically 0 is said to have de-
gree −.
For m a nonnegative integer, let P m (F) denote the set of all poly- 23
nomials with coefficients in F and degree at most m. You should ver-
ify that P m (F) is a subspace of P(F); hence P m (F) is a vector space.
This vector space is finite dimensional because it is spanned by the list
m
k
(1,z,...,z ); here we are slightly abusing notation by letting z denote
a function (so z is a dummy variable).
A vector space that is not finite dimensional is called infinite di- Infinite-dimensional
mensional. For example, P(F) is infinite dimensional. To prove this, vector spaces, which
consider any list of elements of P(F). Let m denote the highest degree we will not mention
of any of the polynomials in the list under consideration (recall that by much anymore, are the
definition a list has finite length). Then every polynomial in the span of center of attention in
this list must have degree at most m. Thus our list cannot span P(F). the branch of
Because no list spans P(F), this vector space is infinite dimensional. mathematics called
The vector space F , consisting of all sequences of elements of F, functional analysis.
∞
is also infinite dimensional, though this is a bit harder to prove. You Functional analysis
should be able to give a proof by using some of the tools we will soon uses tools from both
develop. analysis and algebra.
Suppose v 1 ,...,v m ∈ V and v ∈ span(v 1 ,...,v m ). By the definition
of span, there exist a 1 ,...,a m ∈ F such that

v = a 1 v 1 +· · ·+ a m v m .
Consider the question of whether the choice of a’s in the equation
above is unique. Suppose ˆ a 1 ,..., ˆ a m is another set of scalars such that
v = ˆ a 1 v 1 +· · ·+ ˆ a m v m .

Subtracting the last two equations, we have
0 = (a 1 − ˆ a 1 )v 1 +· · ·+ (a m − ˆ a m )v m .

Thus we have written 0 as a linear combination of (v 1 ,...,v m ). If the
only way to do this is the obvious way (using 0 for all scalars), then
each a j − ˆ a j equals 0, which means that each a j equals ˆ a j (and thus
the choice of a’s was indeed unique). This situation is so important
that we give it a special name—linear independence—which we now
define.
A list (v 1 ,...,v m ) of vectors in V is called linearly independent if
the only choice of a 1 ,...,a m ∈ F that makes a 1 v 1 +···+ a m v m equal
0is a 1 = ··· = a m = 0. For example,