Question

(a) Consider a linear transformation$\mathit{T}$ from ${\mathit{ℝ}}^{\mathbf{5}}$to ${\mathit{ℝ}}^{\mathbf{3}}$. What are the possible values of $\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{k}\mathbf{e}\mathbf{r}\mathbf{T}\mathbf{\right)}$? Explain.

(b) Consider a linear transformation$\mathit{T}$ from ${\mathit{ℝ}}^{\mathbf{4}}$to ${\mathit{ℝ}}^{\mathbf{7}}$. What are the possible values of $\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{i}\mathbf{m}\mathbf{T}\mathbf{\right)}$? Explain.

Short answer

Thus,

(a) the possible values of $\dim Ker\left(T\right)$are 5,4,3,2.

(b) the possible values of$\mathrm{dimIm}\left(T\right)$ are 0,1,2,3,4.

Step-by-step solution

(a) Possible values of dim(kerT).

The rank of a$3\times 5$matrix is bounded by 3, so it is equal to 0,1,2,3. Therefore,$\dim Ker\left(T\right)=5-rk\left(T\right)$ is equal to 5,4,3,2.

Hence, the possible values of $\dim Ker\left(T\right)$are 5,4,3,2.

(b) Possible values ofdim(imT). 

The rank of a matrix is bounded by 4, so it is equal to 0,1,2,3,4. Therefore, $\mathrm{dimIm}\left(T\right)=rk\left(T\right)$ is equal to 0,1,2,3,4.

Hence, the possible values of$\mathrm{dimIm}\left(T\right)$ are 0,1,2,3,4.