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Unformatted text preview: weights Examples of linear combinations of v 1 and v 2 :
3v 1 2v 2 , 1
3 v1, v1 2v 2 , 0 4 EXAMPLE: 2 Let v 1 2 and v 2 . Express 1
2
each of the following as a linear combination of v 1 and v 2 :
0
4
6
7
, b
, c
, d
a
3
1
6
4
x2
8
6
4
2 −8 −6 −4 −2 x1 −2 2 4 6 8 −4
−6
−8 5 1
Let a 1 EXAMPLE: 0 4
, a2 3 3
, a3 2
14 6 , 10 1
and b 8 . 5
Determine if b is a linear combination of a 1 , a 2 , and a 3 .
Solution: Vector b is a linear combination of a 1 , a 2 , and a 3 if can
we find weights x 1 , x 2 , x 3 such that
x 1 a 1 x 2 a 2 x 3 a 3 b.
Vector Equation (fillin): Corresponding System: 3x 1 4x 2 3x 3 1 2x 2 x1 6x 3 8 14x 2 10x 3 5 6 Corresponding Augmented Matrix:
1 4 3 0 2 6 8 3 14 10 100 x 1 ___ 2 Í x 2 ___ 001 5 1 010 1 2 x 3 ___ Review of the last example: a 1 , a 2 , a 3 and b are columns of the
augmented matrix
1 4 3 1 0 2 6 8 3 14 10 5 a1 a2 a3 b
Solution to
x1a1 x2a2 x3a3 b
is found by solving the linear system whose augmented matrix is
a1 a2 a3 b . 7 A vector equation
x1a1 x2a2 xnan b...
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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

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