A First Course in Linear Algebra, 2017a

466 Vector Spaces
Definition 9.12: Linear Combination
Let V beavectorspaceandlet{⃗v 1 ,⃗v 2 ,···,⃗v n }⊆V. A vector ⃗v ∈ V is called a linear combination
of the⃗v i if there exist scalars c i ∈ R such that
⃗v = c 1 ⃗v 1 + c 2 ⃗v 2 + ···+ c n ⃗v n
This definition leads to our next concept of span.
Definition 9.13: Span of Vectors
Let {⃗v 1 ,···,⃗v n }⊆V. Then
span{⃗v 1 ,···,⃗v n } =
{ n∑
c i ⃗v i : c i ∈ R
i=1
}
When we say that a vector ⃗w is in span{⃗v 1 ,···,⃗v n } we mean that ⃗w can be written as a linear combination
of the ⃗v i . We say that a collection of vectors {⃗v 1 ,···,⃗v n } is a spanning set for V if V =
span{⃗v 1 ,···,⃗v n }.
Consider the following example.
Example 9.14: Matrix Span
[ ] [ 1 0 0 1
Let A = , B =
0 2 1 0
]
.DetermineifA and B are in
{[
1 0
span{M 1 ,M 2 } = span
0 0
] [
0 0
,
0 1
]}
Solution.
First consider A. We want to see if scalars s,t can be found such that A = sM 1 +tM 2 .
[ ] [ ] [ ]
1 0 1 0 0 0
= s +t
0 2 0 0 0 1
The solution to this equation is given by
1 = s
2 = t
and it follows that A is in span{M 1 ,M 2 }.
Now consider B. Again we write B = sM 1 +tM 2 and see if a solution can be found for s,t.
[ ] [ ] [ ]
0 1 1 0 0 0
= s +t
1 0 0 0 0 1