Unformatted text preview: has the same solution set as the linear system whose
augmented matrix is
a1 a2 an b . In particular, b can be generated by a linear combination of
a 1 , a 2 , , a n if and only if there is a solution to the linear
system corresponding to the augmented matrix. 8 The Span of a Set of Vectors
EXAMPLE: Let v 4 0
. Label the origin 5
together with v, 2v and 1. 5v on the graph below. 0
0 x3 x2
x1 v, 2v and 1. 5v all lie on the same line.
Span v is the set of all vectors of the form cv.
Here, Span v a line through the origin. 9 EXAMPLE: Label u, v, u v and 3u 4v on the graph below. x3 x2 x1 u, v, u v and 3u 4v all lie in the same plane.
Span u, v is the set of all vectors of the form x 1 u x 2 v.
Here, Span u, v a plane through the origin. 10 Definition
Suppose v 1 , v 2 , , v p are in R n ; then
Span v 1 , v 2 , , v p set of all linear combinations of
v1, v2, , vp.
Stated another way: Span v 1 , v 2 ,
all vectors that can be written as , v p is the collection of x1v1 x2v2 xpvp
where x 1 , x 2 , EXAMPLE: , x p are scalars.
Let v 1 2
1 and v 2 4
2 . (a) Find a vector in Spa...
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