StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q4E

Expert-verifiedFound in: Page 245

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Let A ****be an ${\mathbf{n}}{\mathbf{\times}}{\mathbf{m}}$ ****matrix. Is the formula ${{\left(\mathrm{kerA}\right)}}^{{\mathbf{\perp}}}{\mathbf{=}}{\mathbf{im}}{\left({A}^{T}\right)}$ ****necessarily true? Explain.**

Yes.

Consider a $n\times m$ matrix A.

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

By the definition, $\mathrm{im}{\left({\mathrm{A}}^{\mathrm{T}}\right)}^{\perp}={\mathrm{kerA}}^{{\mathrm{T}}^{\mathrm{T}}}$.

As role="math" localid="1659498837555" ${A}^{{T}^{T}}=A$and ${A}^{{\perp}^{\perp}}=A$ , take $\perp $both side in the equation $im{\left({A}^{T}\right)}^{\perp}=ker{A}^{{T}^{T}}$as follows.

$im{\left({A}^{T}\right)}^{\perp}=ker{A}^{{T}^{T}}\phantom{\rule{0ex}{0ex}}im{\left({A}^{T}\right)}^{\perp}=kerA\phantom{\rule{0ex}{0ex}}{\left\{im{\left({A}^{T}\right)}^{\perp}\right\}}^{\perp}={\left(kerA\right)}^{\perp}\phantom{\rule{0ex}{0ex}}im\left({A}^{T}\right)={\left(kerA\right)}^{\perp}$

Hence, the statement is true

94% of StudySmarter users get better grades.

Sign up for free