Q3.4-2E In Exercises聽聽1聽through 18 , ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q3.4-2E (page 159)

In Exercises $1$ through $18$ , determine whether the vector $\overset{鈫�}{x}$ is in the span $V$ of the vectors ${\overset{鈫�}{v}}_{1}, 鈰�, {\overset{鈫�}{v}}_{m}$ (proceed 鈥渂y inspection鈥� if possible, and use the reduced row-echelon form if necessary). If $\overset{鈫�}{x}$ is in $V$ , find the coordinates of $\overset{鈫�}{x}$ with respect to the basis $I = {\overset{鈫�}{v}}_{1}, 鈰�, {\overset{鈫�}{v}}_{m}$ of $V$ , and write the coordinate vector $[\overset{鈫�}{x}] I$ .
$\overset{鈫�}{x} = [\begin{matrix} 23 \\ 29 \end{matrix}], {\overset{鈫�}{v}}_{1} = [\begin{matrix} 46 \\ 58 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 61 \\ 67 \end{matrix}]$

Short Answer

Expert verified

The coordinates of the vector is, ${\overset{鈫�}{x}}_{I} = [\begin{matrix} \frac{1}{2} \\ 0 \end{matrix}]$ .

Step by step solution

Consider the vectors

The vectors are,

$\overset{鈫�}{x} = [\begin{matrix} 23 \\ 29 \end{matrix}], {\overset{鈫�}{v}}_{1} = [\begin{matrix} 46 \\ 58 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 61 \\ 67 \end{matrix}]$ .

Check for the coordinates of the vector

The formula is, $\overset{鈫�}{x} = c_{1} {\overset{鈫�}{v}}_{1} + c_{2} {\overset{鈫�}{v}}_{2}$ .

$[\begin{matrix} 23 \\ 29 \end{matrix}] = c_{1} [\begin{matrix} 46 \\ 58 \end{matrix}] + c_{2} [\begin{matrix} 61 \\ 67 \end{matrix}] 46 c_{1} + 61 c_{2} = 23, 58 c_{1} + 67 c_{2} = 29 c_{1} = \frac{1}{2}, c_{2} = 0$

Substitute these values in the formula.

$\overset{鈫�}{x} = c_{1} {\overset{鈫�}{v}}_{1} + c_{2} {\overset{鈫�}{v}}_{2} [\begin{matrix} 23 \\ 29 \end{matrix}] = \frac{1}{2} [\begin{matrix} 46 \\ 58 \end{matrix}] + 0 [\begin{matrix} 61 \\ 67 \end{matrix}] {\overset{鈫�}{x}}_{I} = [\begin{matrix} \frac{1}{2} \\ 0 \end{matrix}]$

Final answer

The coordinates of the vector is, ${\overset{鈫�}{x}}_{I} = [\begin{matrix} \frac{1}{2} \\ 0 \end{matrix}]$ .

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91影视

Short Answer

Step by step solution

Consider the vectors

Check for the coordinates of the vector

Final answer

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