Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - Problems - Page 32: 15

Answer

$1v_1 + 0v_2 = \begin{bmatrix} 7 \\ 1 \\ -6 \end{bmatrix}$ $0v_1 + 1v_2 = \begin{bmatrix} -5 \\ 3 \\ 0 \end{bmatrix}$ $1v_1 + 1v_2 = \begin{bmatrix} 2 \\ 4 \\ -6 \end{bmatrix}$ $1v_1 - 1v_2 = \begin{bmatrix} 12 \\ -2 \\ -6 \end{bmatrix}$ $1v_1 - 2v_2 = \begin{bmatrix} 17 \\ -5 \\ -6 \end{bmatrix}$

Work Step by Step

Given $v_1 = \begin{bmatrix} 7 \\ 1 \\ -6 \end{bmatrix}$ and $v_2 = \begin{bmatrix} -5 \\ 3 \\ 0 \end{bmatrix}$, we can generate vectors in the span of $\{v_1, v_2\}$ by creating linear combinations: $1v_1 + 0v_2 = v_1 = \begin{bmatrix} 7 \\ 1 \\ -6 \end{bmatrix}$ $0v_1 + 1v_2 = v_2 = \begin{bmatrix} -5 \\ 3 \\ 0 \end{bmatrix}$ $1v_1 + 1v_2 = \begin{bmatrix} 2 \\ 4 \\ -6 \end{bmatrix}$ $1v_1 - 1v_2 = \begin{bmatrix} 12 \\ -2 \\ -6 \end{bmatrix}$ $1v_1 - 2v_2 = \begin{bmatrix} 17 \\ -5 \\ -6 \end{bmatrix}$ Of course, by selecting different weights, you can generate an infinite number of possible vectors.
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