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Q7E

Expert-verifiedFound in: Page 245

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

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

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

Consider a $n\times m$ matrix A .

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

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

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

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

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