Please answer part b) and c) in particular from section 3.4 of "Linear Algebra with Applications" textbook. In Exercises 19 through 24, find the matrix $ B $ of the linear transformation $ T(\vec{x}) = A\vec{x} $ with respect to the basis $ \mathfrak{B} = (\vec{v}_1, \vec{v}_2) $. For practice, solve each problem in three ways: (a) Use the formula $ B = S^{-1}AS $, (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct $ B $ "column by column."**20.**\[ A = \begin{bmatrix} -3 & 4 \\ 4 & 3 \end{bmatrix} \]\[ \vec{v}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \]\[ \vec{v}_2 = \begin{bmatrix} -2 \\ 1 \end{bmatrix} \]

Consider the given matrix and given vectors.

Answered: In Exercises 19 through 24, find the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Please answer part b) and c) in particular from section 3.4 of "Linear Algebra with Applications" textbook.

In Exercises 19 through 24, find the matrix $ B $ of the linear transformation $ T(\vec{x}) = A\vec{x} $ with respect to the basis $ \mathfrak{B} = (\vec{v}_1, \vec{v}_2) $. For practice, solve each problem in three ways: (a) Use the formula $ B = S^{-1}AS $, (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct $ B $ "column by column."

**20.**

\[ A = \begin{bmatrix} -3 & 4 \\ 4 & 3 \end{bmatrix} \]
\[ \vec{v}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \]
\[ \vec{v}_2 = \begin{bmatrix} -2 \\ 1 \end{bmatrix} \]

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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