Consider the set of vectors )() 1 2 1 S = 2 3 1 -2 (i) Determine if this set of vectors is linearly independent or linearly dependent. (ii) If the set is linearly dependent, determine a linear dependent relationship between these vectors. (iii) Consider the transformation xi + 2x2 + x3 2x1 + 3x2 + x3 + 2 T 13 3x1 + x2 – 2x3 - Determine if T is a linear transformation or not.

Consider the set of vectors )() 1 2 1 S = 2 3 1 -2 (i) Determine if this set of vectors is linearly independent or linearly dependent. (ii) If the set is linearly dependent, determine a linear dependent relationship between these vectors. (iii) Consider the transformation xi + 2x2 + x3 2x1 + 3x2 + x3 + 2 T 13 3x1 + x2 – 2x3 - Determine if T is a linear transformation or not.

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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Question

**Problem 5: Vector and Transformation Analysis**

Consider the set of vectors:

\[
S = \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix} \right\}
\]

(i) Determine if this set of vectors is linearly independent or linearly dependent.

(ii) If the set is linearly dependent, determine a linear dependent relationship between these vectors.

(iii) Consider the transformation:

\[
T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 + 2x_2 + x_3 \\ 2x_1 + 3x_2 + x_3 + 2 \\ 3x_1 + x_2 - 2x_3 \end{pmatrix}
\]

Determine if $ T $ is a linear transformation or not.

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