Q3.4-17E In Exercises聽1 through 18, dete... [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 1: Q3.4-17E (page 1)

In Exercises1 through 18, determine whether the vector $\overset{鈫�}{x}$ is in the span V of the vectors $\overset{鈫�}{v_{1}},,,,, \overset{鈫�}{v_{m}}$ (proceed 鈥渂y inspection鈥� if possible, and use the reduced row-echelon form if necessary). If localid="1664192105355" $\overset{鈬赌}{x}$ is in V, find the coordinates of $\overset{鈬赌}{x}$ with respect to the basis $I = \overset{鈬赌}{v_{1}}, . . . . \overset{鈬赌}{v_{m}}$ of V, and write the coordinate vector $[\overset{鈫�}{x}] I$ .
localid="1664193284963" $\overset{鈫�}{x} = [\begin{matrix} 1 \\ 1 \\ 1 \\ - 1 \end{matrix}]; \overset{鈫�}{v_{1}} = [\begin{matrix} 1 \\ 0 \\ 2 \\ 0 \end{matrix}]; \overset{鈫�}{v_{2}} = [\begin{matrix} 0 \\ 1 \\ 3 \\ 0 \end{matrix}]; \overset{鈫�}{v_{3}} = [\begin{matrix} 0 \\ 0 \\ 4 \\ 1 \end{matrix}]$

Short Answer

Expert verified

The coordinates of the vector is, $\overset{鈬赌}{x} I = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}]$ .

Step by step solution

Consider the vectors.

The vectors are,

$\overset{鈫�}{x} = [\begin{matrix} 1 \\ 1 \\ 1 \\ - 1 \end{matrix}]; \overset{鈫�}{v_{1}} = [\begin{matrix} 1 \\ 0 \\ 2 \\ 0 \end{matrix}]; \overset{鈫�}{v_{2}} = [\begin{matrix} 0 \\ 1 \\ 3 \\ 0 \end{matrix}]; \overset{鈫�}{v_{3}} = [\begin{matrix} 0 \\ 0 \\ 4 \\ 1 \end{matrix}]$

Check for the coordinates of the vector.

The formula is, $\overset{鈬赌}{x} = c_{1} \overset{鈬赌}{v_{1}} + c_{2} \overset{鈬赌}{v_{2}}$ .

By inspection method,

$[\begin{matrix} 1 \\ 0 \\ 2 \\ 0 \end{matrix}] + [\begin{matrix} 0 \\ 1 \\ 3 \\ 0 \end{matrix}] - [\begin{matrix} 0 \\ 0 \\ 4 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ 1 \\ - 1 \end{matrix}]$

Substitute these values in the formula.

role="math" localid="1664193247857" $\overset{鈬赌}{x} = c_{1} \overset{鈬赌}{v_{1}} + c_{2} \overset{鈬赌}{v_{2}} 鈭� \overset{鈬赌}{x} I = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}]$

Final answer

The coordinates of the vector is, $\overset{鈬赌}{x} I = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}]$ .

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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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