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Math 70 sec1_3ov

7 a vector equation x1a1 x2a2 xnan b has the same

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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...
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