Question

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

Short answer

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

Step-by-step solution

Determine the relation.

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)=${\left(kerB\right)}^{\perp }$.

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$im\left(A\right)and{\left(kerA\right)}^{\perp }isim\left(A\right)={\left(kerA\right)}^{\perp }$.