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1. Write a statement involving the solutions of a vector equation that's equivalent to each claim: 1. "The set of vectors 1 11 0 0 " 1 0 0 0 5 -5 4 " -3 8 -5 4 " -2 17 -15 12 spans

1. Write a statement involving the solutions of a vector equation that's equivalent to each claim: 1. "The set of vectors 1 11 0 0 " 1 0 0 0 5 -5 4 " -3 8 -5 4 " -2 17 -15 12 spans

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**1. Write a statement involving the solutions of a vector equation that's equivalent to each claim:** 1. "The set of vectors \[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -3 \\ 8 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -2 \\ 17 \\ -15 \\ 12 \\ \end{bmatrix} \right\} \] spans \(\mathbb{R}^4\)." 2. "The set of vectors \[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -3 \\ 8 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -2 \\ 17 \\ -15 \\ 12 \\ \end{bmatrix} \right\} \] does not span \(\mathbb{R}^4\)." **2. Explain how to determine which of these statements is true.** To determine which statement is true, set up a matrix with the vectors as columns and row-reduce to echelon form. Check if there are four pivot points. If there are, the vectors span \(\mathbb{R}
Transcribed Image Text:**1. Write a statement involving the solutions of a vector equation that's equivalent to each claim:** 1. "The set of vectors \[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -3 \\ 8 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -2 \\ 17 \\ -15 \\ 12 \\ \end{bmatrix} \right\} \] spans \(\mathbb{R}^4\)." 2. "The set of vectors \[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 5 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -3 \\ 8 \\ -5 \\ 4 \\ \end{bmatrix}, \begin{bmatrix} -2 \\ 17 \\ -15 \\ 12 \\ \end{bmatrix} \right\} \] does not span \(\mathbb{R}^4\)." **2. Explain how to determine which of these statements is true.** To determine which statement is true, set up a matrix with the vectors as columns and row-reduce to echelon form. Check if there are four pivot points. If there are, the vectors span \(\mathbb{R}
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