This preview shows page 1. Sign up to view the full content.
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
3
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...
View
Full
Document
This note was uploaded on 02/12/2014 for the course MATH 70 taught by Professor Mcgrath during the Spring '13 term at Tufts.
 Spring '13
 MCGRATH
 Math, Linear Algebra, Algebra, Equations, Vectors

Click to edit the document details