Linear algebra review number 1 **Linear Algebra Homework Questions**---**Instructions:**- Carefully read each question and provide detailed responses.- Use proper mathematical notation and set theories.- Show all steps in your workings for full marks.---1. **Problem 1:** If $ T $ is a linear transformation with $ T:\mathbb{R}^2 \to \mathbb{R}^2 $ and \[ T \left[ \begin{array}{c} 1 \\ 2 \end{array} \right] = \left[ \begin{array}{c} 5 \\ 2 \end{array} \right] \quad \text{and} \quad T \left[ \begin{array}{c} -3 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 3 \\ 1 \end{array} \right], \] find the standard matrix for the linear transformation $ T $.2. **Problem 2:** Give the solution set of the equation $ x - 2y + 3z = 12 $ using proper set notation.3. **Problem 3:** Are the vectors \[ \left[ \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right], \quad \left[ \begin{array}{c} 2 \\ 3 \\ 1 \end{array} \right], \quad \left[ \begin{array}{c} 2 \\ 5 \\ 2 \end{array} \right] \] linearly independent? If not, then write a dependency relation.4. **Problem 4:** $ T $ is a linear transformation with $ T: \mathbb{R}^3 \to \mathbb{R}^3 $ defined by \[ T \left( \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right] \right) = \left[ \begin{array}{c} x_1 + x_2 - x_3 \\ x_1 - x_2 + x_3 \\ x_1 + x_3 \end{array} \right]. \]

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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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Linear algebra review number 1

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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