Page 36 - Linear Algebra Done Right

Chapter 2. Finite-Dimensional Vector Spaces
22
Span and Linear Independence
A linear combination of a list (v 1 ,...,v m ) of vectors in V is a vector
of the form
2.1 a 1 v 1 +· · ·+ a m v m ,
where a 1 ,...,a m ∈ F. The set of all linear combinations of (v 1 ,...,v m )
Some mathematicians is called the span of (v 1 ,...,v m ), denoted span(v 1 ,...,v m ). In other
use the term linear words,
span, which means the
same as span. span(v 1 ,...,v m ) ={a 1 v 1 + ··· + a m v m : a 1 ,...,a m ∈ F}.
3
As an example of these concepts, suppose V = F . The vector

(7, 2, 9) is a linear combination of (2, 1, 3), (1, 0, 1) because
(7, 2, 9) = 2(2, 1, 3) + 3(1, 0, 1).

Thus (7, 2, 9) ∈ span (2, 1, 3), (1, 0, 1) .
You should verify that the span of any list of vectors in V is a sub-
space of V. To be consistent, we declare that the span of the empty list
() equals {0} (recall that the empty set is not a subspace of V).
If (v 1 ,...,v m ) is a list of vectors in V, then each v j is a linear com-
bination of (v 1 ,...,v m ) (to show this, set a j = 1 and let the other a’s
in 2.1 equal 0). Thus span(v 1 ,...,v m ) contains each v j . Conversely,
because subspaces are closed under scalar multiplication and addition,
every subspace of V containing each v j must contain span(v 1 ,...,v m ).
Thus the span of a list of vectors in V is the smallest subspace of V
containing all the vectors in the list.
If span(v 1 ,...,v m ) equals V, we say that (v 1 ,...,v m ) spans V.A
Recall that by vector space is called finite dimensional if some list of vectors in it
n
definition every list has spans the space. For example, F is finite dimensional because
finite length.
(1, 0,..., 0), (0, 1, 0,..., 0),...,(0,..., 0, 1)

n
spans F , as you should verify.
Before giving the next example of a finite-dimensional vector space,
we need to define the degree of a polynomial. A polynomial p ∈P(F)
is said to have degree m if there exist scalars a 0 ,a 1 ,...,a m ∈ F with
a m = 0 such that

2.2 p(z) = a 0 + a 1 z + ··· + a m z m