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Expert-verified Found in: Page 442 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Find all the eigenvalues and “eigenvectors” of the linear transformations.${\mathbf{T}}\left(f\right){\mathbf{=}}{\mathbf{f}}{\mathbf{\text{'}}}{\mathbf{‐}}{\mathbf{f}}{\mathbf{}}{\mathbf{from}}{\mathbf{}}{\mathbf{C}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{}}{\mathbf{to}}{\mathbf{}}{\mathbf{C}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}$

The eigenvalues and eigenvectors of the linear transformation is $\mathrm{\lambda }\in {\mathrm{R}}_{,}{\mathrm{E}}_{\mathrm{\lambda }}=\mathrm{span}\left({\mathrm{e}}^{\left(\mathrm{\lambda }+1\right)\mathrm{t}}\right)$.

See the step by step solution

## Step 1: Define eigenvalues

The scalar values that are associated with the vectors of the linear equations in the matrix are called eigenvalues.

${\mathbf{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{=}}{\mathbf{\lambda }}\stackrel{\mathbf{\to }}{\mathbf{x}}$, here $\stackrel{\mathbf{\to }}{\mathbf{x}}$ is eigenvector and ${\mathbf{\lambda }}$ is the eigenvalue.

## Step 2: Use the formula and find the eigenvalues and eigenvectors

Consider the given equation, $\mathrm{T}\left(\mathrm{f}\right)=\mathrm{f}\text{'}-\mathrm{f}$

Solve,

$\mathrm{T}\left(\mathrm{f}\right)=\mathrm{\lambda f}$

Substitute the value of $\mathrm{T}\left(\mathrm{f}\right)=\mathrm{\lambda f}$

$\mathrm{f}\text{'}-\mathrm{f}=\mathrm{\lambda f}\phantom{\rule{0ex}{0ex}}\mathrm{f}\text{'}=\left(\mathrm{\lambda }+1\right)\mathrm{f}\phantom{\rule{0ex}{0ex}}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{C}·{\mathrm{e}}^{\left(\mathrm{\lambda }+1\right)\mathrm{x}},\mathrm{C}\in \mathrm{R}$

Hence, for the eigenvalue $\mathrm{\lambda }\in \mathrm{R}$ , the eigenvector space is ${\mathrm{E}}_{\mathrm{\lambda }}=\mathrm{span}\left({\mathrm{e}}^{\left(\mathrm{\lambda }+1\right)\mathrm{x}}\right)$

Thus, $\mathrm{\lambda }\in \mathrm{R},{\mathrm{E}}_{\mathrm{\lambda }}=\mathrm{span}\left({\mathrm{e}}^{\left(\mathrm{\lambda }+1\right)\mathrm{t}}\right)$ ### Want to see more solutions like these? 