Question

For two nonparallel vectors$\overrightarrow{\mathbf{v}}$and$\overrightarrow{\mathbf{w}}$in${\mathit{R}}^{\mathbf{3}}$ , consider the linear transformationrole="math" localid="1660626507967" $\mathit{T}\mathbf{\left(}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{\left[}\begin{array}{ccc}\overrightarrow{\mathbf{v}}& \overrightarrow{\mathbf{w}}& \overrightarrow{\mathbf{x}}\end{array}\mathbf{\right]}$ from ${\mathit{R}}^{\mathbf{3}}$to $\mathbf{ℝ}$. Describe the kernel ofTgeometrically. What is the image of T ?

Short answer

Therefore, the kernel geometric and image of T is given by:

$$\begin{gathered} \ker \left(T\right)=span(\overrightarrow{v},\overrightarrow{w}) \\ im\left(T\right)=\mathrm{ℝ} \end{gathered}$$

Step-by-step solution

Given

Consider, the linear transformation is given by,

$T\left(\overrightarrow{\mathrm{x}}\right)=det\left[\begin{array}{ccc}\overrightarrow{\mathrm{v}}& \overrightarrow{\mathrm{w}}& \overrightarrow{\mathrm{x}}\end{array}\right]$

To find the kernel geometric and image

The kernel ofconsists of all the vectors$\overrightarrow{x}\in {\mathrm{ℝ}}^3$such that$T\left(\overrightarrow{x}\right)=0$, which gives us

$\text{det}\left[\begin{array}{ccc}\overrightarrow{v}& \overrightarrow{w}& \overrightarrow{x}\end{array}\right]\text{=0}$

Given that$\overrightarrow{v}$and$\overrightarrow{w}$are non-parallel vectors, this matrix being non-invertible can only be accomplished by$\overrightarrow{x}$being a linear combination $\overrightarrow{v}$ofand$\overrightarrow{w}$. Therefore,

$\text{ker}\left(T\right)=\text{span}(\overrightarrow{v},\overrightarrow{w})$

If$\overrightarrow{x}\notin \text{span}(\overrightarrow{v},\overrightarrow{w})$, then the given matrix is invertible.

So its determinant is different than 0.

Given that the image of T must be a subspace of, we have$\text{im}\left(T\right)=\mathrm{ℝ}$.

Therefore,

$\begin{array}{rcl}\text{ker}\left(T\right)& =& \text{span}(\overrightarrow{v},\overrightarrow{w})\\ \text{im}\left(T\right)& =& \mathrm{ℝ}\\ & & \end{array}$