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Q29E

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Found in: Page 393

### Linear Algebra With Applications

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

# Consider a symmetric matrix A. If the vector $\stackrel{\mathbf{⇀}}{\mathbf{v}}$ is in the image of A and $\stackrel{\mathbf{⇀}}{\mathbit{w}}$ is in the kernel of A, is $\stackrel{\mathbf{⇀}}{\mathbf{v}}$ necessarily orthogonal to $\stackrel{\mathbf{⇀}}{w}$? Justify your answer.

The $\stackrel{⇀}{\mathrm{v}}$ is orthogonal to $\stackrel{⇀}{w}$.

See the step by step solution

## Step 1: The matrix

• A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
• For example, is a two-row, three-column matrix

## Step 2: Determine the orthogonal matrix

A theorem we studied in this book states that

${\left(\mathrm{im}\mathrm{A}\right)}^{-}=\mathrm{ker}\left({\mathrm{A}}^{\mathrm{T}}\right)$

And we know that A is symmetric $⇄\mathrm{A}={\mathrm{A}}^{\mathrm{T}}$, which implies

${\left(\mathrm{im}\mathrm{A}\right)}^{-}=\mathrm{ker}\left({\mathrm{A}}^{\mathrm{T}}\right)\to {\left(\mathrm{im}\mathrm{A}\right)}^{-}=\mathrm{ker}\left(\mathrm{A}\right)$

Hence $\stackrel{⇀}{v}$ is orthogonal to $\stackrel{⇀}{\mathrm{w}}$.