Question

Let Abe 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.

Short answer

Yes.

Step-by-step solution

Determine the property of orthogonal.

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.

$$\begin{gathered} im{\left(A^T\right)}^{\perp }=ker{A}^{T^T} \\ im{\left(A^T\right)}^{\perp }=kerA \\ {\left\{im{\left(A^T\right)}^{\perp }\right\}}^{\perp }={\left(kerA\right)}^{\perp } \\ im\left(A^T\right)={\left(kerA\right)}^{\perp } \end{gathered}$$

Hence, the statement is true