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Q16E

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Found in: Page 323

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.Rotation through an angle of ${{\mathbf{180}}}^{{\mathbf{°}}}$ in ${{\mathbit{R}}}^{{\mathbf{2}}}$.

So, A is diagonalizable.

See the step by step solution

## Step 1: Definition of diagonalizable

A matrix is known as diagonalizable if it can be written in the form of ${\mathbit{A}}{\mathbf{=}}{\mathbit{P}}{\mathbit{D}}{{\mathbit{P}}}^{\mathbf{-}\mathbf{1}}$.

## Step 2: Checking whether the matrix diagonalizable

Rotation through an angle of ${180}^{°}$ will turn any vector v into -v , so role="math" localid="1659525298008" $A=-{l}_{2}$, and all vectors $v\in {R}^{2}$are eigenvectors of A with the corresponding eigenvalue being $\lambda =-1$. The very fact that $A=-{l}_{2}$ is already diagonal proves that it's diagonalizable.