1.4 - 1.4 Matrix Equation Ax = b Key idea: View linear...

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1.4 Matrix Equation A~x = ~ b Key idea: View linear combinations of vectors as a matrix-vector product: A~x = h ~ a 1 ~ a 2 ··· ~ a n i x 1 x 2 . . . x n = x 1 ~ a 1 + x 2 ~ a 2 + ··· + x n ~ a n where A is an m × n matrix. EX: 1 0 2 3 - 1 3 2 - 1 1 1 0 1 1 0 1 - 2 = 1 · 1 - 1 1 + 0 · 0 3 1 + 1 · 2 2 0 - 2 · 3 - 1 1 = 1 - 1 1 + 2 2 0 + - 6 2 - 2 = - 3 3 - 1 1
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Properties: For any m × n matrix A , A ( ~u + ~v ) = A~u + A~v , ~u , ~v ∈ R n A ( c~u ) = c ( A~u ) , c ∈ R We’ll use these rules from now on! Review: Have 3 EQUIVALENT ways of writ- ing a linear system of equations: · A ~x = ~ b Am × n · h ~ a 1 | ~ a 2 | ··· | ~ a n i ~x = ~ b · x 1 ~ a 1 + x 2 ~ a 2 + ··· + x n ~ a n = ~ b They all have the same soln set Q: When is there a soln ~x ? A: The answer is given below. Theorem 4: Let
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1.4 - 1.4 Matrix Equation Ax = b Key idea: View linear...

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