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Determine (by justifying) whether the following set of vectors are linearly dependent or linearly independent. (a) { 卧 -2 -6 (b) { -1 7 14 2 (c) {

Determine (by justifying) whether the following set of vectors are linearly dependent or linearly independent. (a) { 卧 -2 -6 (b) { -1 7 14 2 (c) {

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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### Problem 5: Linear Dependence of Vectors Determine (by justifying) whether the following set of vectors are linearly dependent or linearly independent. #### Set (a): Vectors: \[ \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 5 \\ 2 \\ \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \] #### Set (b): Vectors: \[ \begin{bmatrix} -2 \\ 2 \\ 3 \\ 7 \\ \end{bmatrix}, \begin{bmatrix} -6 \\ 1 \\ -1 \\ 14 \\ \end{bmatrix} \] #### Set (c): Vectors: \[ \begin{bmatrix} 2 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ -1 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 3 \\ 2 \\ \end{bmatrix} \] ### Guidelines: For each set of vectors, analyze whether they are linearly dependent or independent by constructing the corresponding matrix and reducing it to its row echelon form (REF) or reduced row echelon form (RREF). The condition for linear dependence is that there exists a non-trivial solution to the homogeneous equation \(Ax = 0\); otherwise, the vectors are independent.
Transcribed Image Text:### Problem 5: Linear Dependence of Vectors Determine (by justifying) whether the following set of vectors are linearly dependent or linearly independent. #### Set (a): Vectors: \[ \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 5 \\ 2 \\ \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \] #### Set (b): Vectors: \[ \begin{bmatrix} -2 \\ 2 \\ 3 \\ 7 \\ \end{bmatrix}, \begin{bmatrix} -6 \\ 1 \\ -1 \\ 14 \\ \end{bmatrix} \] #### Set (c): Vectors: \[ \begin{bmatrix} 2 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ -1 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 3 \\ 2 \\ \end{bmatrix} \] ### Guidelines: For each set of vectors, analyze whether they are linearly dependent or independent by constructing the corresponding matrix and reducing it to its row echelon form (REF) or reduced row echelon form (RREF). The condition for linear dependence is that there exists a non-trivial solution to the homogeneous equation \(Ax = 0\); otherwise, the vectors are independent.
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