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Found in: Page 245

### Linear Algebra With Applications

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

# Consider a symmetric ${\mathbf{n}}{\mathbf{×}}{\mathbf{m}}$ matrix A. What is the relationship between Im(A) and ker(A) ?

The relation between $\mathrm{im}\left(\mathrm{A}\right)\mathrm{and}{\left(\mathrm{kerA}\right)}^{\perp }\mathrm{is}\mathrm{im}\left(\mathrm{A}\right)={\left(\mathrm{kerA}\right)}^{\perp }$.

See the step by step solution

## Step 1: Determine the relation.

Consider a $n×m$ matrix A .

Theorem: For any matrix B ,${\mathrm{im}}\left(\mathrm{B}\right){=}{\left(\mathrm{kerB}\right)}^{{\perp }}$.

By the theorem, im(A)=${\left(kerB\right)}^{\perp }$.

## Step 2: Draw the graph of relation between Im(A) and ker (A) .

Sketch the graph to show the relation between $im\left(A\right)and{\left(kerA\right)}^{\perp }$

Hence, the relation between $im\left(A\right)and{\left(kerA\right)}^{\perp }isim\left(A\right)={\left(kerA\right)}^{\perp }$.