91Ó°ÊÓ

Consider two linear spaces V and W . A function T is said to be linear transformation if the following holds.

$$\begin{gathered} \mathbf{T}\mathbf{(}\mathbf{f}\mathbf{+}\mathbf{g}\mathbf{)}\mathbf{=}\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{+}\mathbf{T}\mathbf{\left(}\mathbf{g}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{T}\mathbf{\left(}\mathbf{kf}\mathbf{\right)}\mathbf{=}\mathbf{kT}\mathbf{\left(}\mathbf{f}\mathbf{\right)} \end{gathered}$$

For all elements f,g of V and k is scalar.

An invertible linear transformation is called an isomorphism.

Let’s define a transformation as follows.

$\mathbf{T}\mathbf{:}{\mathbf{R}}^{\mathbf{2}\mathbf{×}\mathbf{2}}\mathbf{→}{\mathbf{R}}^{\mathbf{2}\mathbf{×}\mathbf{2}}\mathbf{}\mathbf{with}\mathbf{}\mathbf{T}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}{\mathbf{P}}^{\mathbf{-}\mathbf{1}}\mathbf{AP}\mathbf{}\mathbf{where}\mathbf{}\mathbf{P}\mathbf{=}\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$.

The given transformation as follows.

$\mathrm{T}\left(\mathrm{M}\right)={\mathrm{P}}^{-1}\mathrm{MP},\mathrm{where}\mathrm{P}=\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]\mathrm{from}{\mathrm{R}}^{2×2}\mathrm{to}{\mathrm{R}}^{2×2}$

By using the definition of linear transformation as follows.

role="math" localid="1659416314032" $$\begin{gathered} \mathrm{T}(\mathrm{A}+\mathrm{B})=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right) \\ \mathrm{T}\left(\mathrm{KA}\right)=\mathrm{kT}\left(\mathrm{A}\right) \end{gathered}$$

Now, to check the first condition as follows.

Let A and B be arbitrary matrices from ${\mathrm{R}}^{2×2}$and as follows.

$$\begin{gathered} \mathrm{T}(\mathrm{A}+\mathrm{B})=P^{-1}(A+B)P \\ =P^{-1}AP+P^{-1}BP \\ \mathrm{T}(\mathrm{A}+\mathrm{B})=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right) \end{gathered}$$

Similarly, to check the second condition as follows.

Let $\mathrm{Î\pm }$be an arbitrary scalar, and $\mathrm{A}∈{\mathrm{R}}^{2×2}$as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{Î\pm ´¡}\right)={\mathrm{P}}^{-1}\left(\mathrm{Î\pm ´¡}\right)\mathrm{P} \\ ={\mathrm{Î\pm \pm Ê}}^{-1}\mathrm{AP} \\ \mathrm{T}\left(\mathrm{Î\pm ´¡}\right)=\mathrm{Î\pm °Õ}\left(\mathrm{A}\right) \end{gathered}$$

Thus, T is a linear transformation.

Next to show that T is an isomorphism is to exhibits the inverse as follows.

Solve T (A) =B for A as follows.

$$\begin{gathered} {\mathrm{PAP}}^{-1}=\mathrm{B} \\ \mathrm{A}={\mathrm{P}}^{-1}\mathrm{Bp} \end{gathered}$$

Since, form the matrix as follows.

$$\begin{gathered} 2×7-5×3=14-15 \\ =-1 \end{gathered}$$

So, ${\mathrm{P}}^{-1}$exist and $\mathrm{A}={\mathrm{P}}^{-1}\mathrm{BP}$is defined.

This shows that the inverse transformation is ${\mathrm{T}}^{-1}={\mathrm{P}}^{-1}\mathrm{BP}$.

Therefore, T is an isomorphism.

Hence, T is a linear transformation and is an isomorphism.