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# notes1-4 - b Continuing the example do the columns of A...

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SECTION 1.4 MATRIX-VECTOR EQUATIONS When the matrix A and the vector x are the right sizes, then the product A x is the linear combination of the columns of A using the corresponding entries of x as weights. The matrix A must have as many columns as the vector x has rows. EXAMPLES. Write the following matrix equation as a vector equation. Then write the system of equations that we need to solve to ﬁnd the unknown vector x . 1 3 - 2 0 5 2 - 3 - 9 6 x 1 x 2 x 3 = 16 13 - 48

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Write the following vector equation as a matrix equation A x = b . x 1 1 2 3 + x 2 1 1 6 = 1 3 0 . For the matrix A in this example, does A x = b have a solution for every
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Unformatted text preview: b ? Continuing the example, do the columns of A span R 3 ? EXAMPLE. Suppose A is a 4 × 3 matrix and b is a vector in R 4 with the property that A x = b has a unique solution. What can you say about the reduced echelon form of A ? NOTE. If the product A x is deﬁned, then the i th entry of the product is the sum of the products of the corresponding entries from the i th row of A and from the vector x . Let’s practice. 1 3-2 5 2-3-9 6 5 3-1 HOMEWORK: SECTION 1.4...
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notes1-4 - b Continuing the example do the columns of A...

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