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Consider the vectors u = k= 6 B - Calculator 5 v= 2, and = 5 1 2 What value of k will make the set {u, v, w} linearly dependent? You may submit your answer as a fraction, if necessary. k makes the set linearly dependent.

Consider the vectors u = k= 6 B - Calculator 5 v= 2, and = 5 1 2 What value of k will make the set {u, v, w} linearly dependent? You may submit your answer as a fraction, if necessary. k makes the set linearly dependent.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Vectors and Linear Dependence** Consider the vectors: \[ \mathbf{u} = \begin{bmatrix} 6 \\ 5 \\ -1 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 7 \\ 2 \\ -5 \end{bmatrix}, \quad \text{and} \quad \mathbf{w} = \begin{bmatrix} \frac{2}{3} \\ 17 \\ k \end{bmatrix} \] **Problem Statement:** What value of \( k \) will make the set \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) linearly dependent? You may submit your answer as a fraction, if necessary. **Interactive Element:** - \( k = \) [input box] makes the set linearly dependent. \[ \text{[Calculator Button]} \] \[ \text{[Check Answer Button]} \] **Explanation:** For the set of vectors to be linearly dependent, there must be a non-trivial solution to the equation \( a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0} \), where not all coefficients \( a, b, c \) are zero.
Transcribed Image Text:**Vectors and Linear Dependence** Consider the vectors: \[ \mathbf{u} = \begin{bmatrix} 6 \\ 5 \\ -1 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 7 \\ 2 \\ -5 \end{bmatrix}, \quad \text{and} \quad \mathbf{w} = \begin{bmatrix} \frac{2}{3} \\ 17 \\ k \end{bmatrix} \] **Problem Statement:** What value of \( k \) will make the set \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) linearly dependent? You may submit your answer as a fraction, if necessary. **Interactive Element:** - \( k = \) [input box] makes the set linearly dependent. \[ \text{[Calculator Button]} \] \[ \text{[Check Answer Button]} \] **Explanation:** For the set of vectors to be linearly dependent, there must be a non-trivial solution to the equation \( a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0} \), where not all coefficients \( a, b, c \) are zero.
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u=\begin{pmatrix}6\\ 5\\ -1\end{pmatrix},\:\:\:v=\begin{pmatrix}7\\ 2\\ -5\end{pmatrix},\:\:\:w=\begin{pmatrix}\frac{2}{3}\\ \frac{17}{3}\\ k\end{pmatrix}

write all 3 vectors in the matrix A

A=\begin{pmatrix}6&7&\frac{2}{3}\\ 5&2&\frac{17}{3}\\ -1&-5&k\end{pmatrix}

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