Math 70 sec1_3ov

v p using weights examples of linear combinations of

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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 (fill-in): 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.

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