Question

Is the transformation$\mathit{L}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}{\mathit{A}}^{\mathbf{T}}$ from ${\mathbf{ℝ}}^{\mathbf{2}\mathbf{\times }\mathbf{3}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathbf{ℝ}}^{\mathbf{3}\mathbf{\times }\mathbf{2}}$ linear? Is Lan isomorphism?

Short answer

The transformation $L\left(A\right)=A^T$is linear and an isomorphism.

Step-by-step solution

Determine the linearity of .

Consider the transformation $L\left(A\right)=A^T$from ${\mathrm{ℝ}}^{2\times 3}\mathrm{to}{\mathrm{ℝ}}^{3\times 2}$to .

A function Dis called a linear transformation on $\mathbf{ℝ}$if the function D satisfies the following properties.

$$\begin{gathered} D(x+y)=D\left(x\right)+D\left(y\right)\mathrm{for}\mathrm{all}x,y\in \mathrm{ℝ}. \\ D\left(\alpha x\right)=\alpha D\left(x\right)\mathrm{for}\mathrm{all}\mathrm{constant}\alpha \in \mathrm{ℝ}. \end{gathered}$$

An invertible linear transformation is called isomorphism or dimension of domain and co-domain is not same then the function is not isomorphism.

Assume $A,B\in {\mathrm{ℝ}}^{2\times 3}\mathrm{then}L\left(A\right)=A^T\mathrm{and}L\left(B\right)=B^T$.

Substitute the value $A^{T}forL\left(A\right)\mathrm{and}L\left(B\right)\mathrm{in}L\left(A\right)+L\left(B\right)$as follows.

$L\left(A\right)+L\left(B\right)=A^T+B^T$

Now, simplify L (A+B) as follows.

role="math" localid="1660116731866" $$\begin{gathered} L(A+B)={(A+B)}^T \\ ={\left(A\right)}^T+{\left(B\right)}^T \\ L(A+B)=L\left(A\right)+L\left(B\right) \end{gathered}$$

Assume $A\in {\mathrm{ℝ}}^{2\times 3}\mathrm{and}\alpha \in \mathrm{ℝ}\mathrm{then}L\left(\alpha A\right)={\left(\alpha A\right)}^T$.

Simplify the equation $L\left(\alpha A\right)={\left(\alpha A\right)}^T$as follows.

role="math" localid="1660116927053" $$\begin{gathered} L\left(\alpha A\right)={\left(\alpha A\right)}^T \\ =\alpha {\left(A\right)}^T \\ L\left(\alpha A\right)=\alpha L\left(A\right) \end{gathered}$$

As $L(A+B)=L\left(A\right)+L\left(B\right)\mathrm{and}L\left(\alpha A\right)=\alpha L\left(A\right)$, by the definition of linear transformation is linear from ${\mathrm{ℝ}}^{2\times 3}\mathrm{to}{\mathrm{ℝ}}^{3\times 2}$.

Determine the isomorphism.

Theorem: Consider a linear transformation T defined from $T:V\to W$then the transformation T is an isomorphism if $ker\left(T\right)=0$and only if where$ker\left(T\right)=\left\{f\left(x\right)\in P:T\left\{f\left(x\right)\right\}=0\right\}$.

Assume $A\in {\mathrm{ℝ}}^{2\times 3}$such that L (A) = 0.

Compare the equations $L\left(A\right)=0\mathrm{and}L\left(A\right)=A^T$and as follows.

$A^T=0$

Take the transpose both side in the equation $A^T=0$as follows.

$$\begin{gathered} A^T=0 \\ {\left(A^T\right)}^T=0^T \\ A=0 \end{gathered}$$

By the definition of kernel, the value (L) = {0}.

As the dimension of kernel(L) is 0, by the theorem the function L is an isomorphism.

Hence, the transformation L is linear and an isomorphism.