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

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

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

An invertible linear transformation is called an isomorphism.

Let鈥檚 define a transformation as follows.

$\mathbf{T}\mathbf{:}{\mathbf{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{鈫�}{\mathbf{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{with}\mathbf{}\mathbf{T}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{a}\mathbf{\left[}\begin{array}{cc}\mathbf{2}& \mathbf{3}\\ \mathbf{4}& \mathbf{5}\end{array}\mathbf{\right]}$

The given transformation as follows.

$\mathrm{T}\left(\mathrm{M}\right)=\mathrm{c}\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],\mathrm{from}\mathrm{R}\mathrm{to}{\mathrm{R}}^{2脳2}$

By using the definition of linear transformation as follows.

$$\begin{gathered} \mathrm{T}(\mathrm{A}+\mathrm{B})=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right) \\ \mathrm{T}(\mathrm{k}\mathrm{A})=\mathrm{kT}\left(\mathrm{A}\right) \end{gathered}$$

Now, to check the first condition as follows.

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

$$\begin{gathered} \mathrm{T}(\mathrm{a}+\mathrm{b})=(\mathrm{a}+\mathrm{b})\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right] \\ =\mathrm{a}\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]+\mathrm{b}\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right] \\ \mathrm{T}(\mathrm{a}+\mathrm{b})=\mathrm{T}\left(\mathrm{a}\right)+\mathrm{T}\left(\mathrm{b}\right) \end{gathered}$$

Similarly, to check the second condition as follows.

Let be an arbitrary scalar, and$\mathrm{A}鈭�{\mathrm{R}}^{2脳2}$ as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{伪补}\right)=\mathrm{伪补}\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right] \\ \mathrm{T}\left(\mathrm{伪补}\right)=\mathrm{伪罢}\left(\mathrm{a}\right) \end{gathered}$$

Thus, T is a linear transformation.

A linear transformation $\mathbf{T}\mathbf{:}\mathbf{V}\mathbf{鈫�}\mathbf{W}$is isomorphism if and only if $\mathbf{ker}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathbf{0}\mathbf{\right\}}$and $\mathbf{lm}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{W}$

Now, check if ker(t) = {0} as follows.

$\ker \left(\mathrm{T}\right)=\left\{\mathrm{A}鈭�{\mathrm{R}}^{2脳2}\left|\mathrm{T}\right(\mathrm{A})=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\right\}$

The next equation as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{a}\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \mathrm{a}\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}2\mathrm{a}& 3\mathrm{a}\\ 4\mathrm{a}& 5\mathrm{a}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$

Equating the corresponding entries as follows.

$$\begin{gathered} 2\mathrm{a}=0 \\ 3\mathrm{a}=0 \\ 4\mathrm{a}=0 \\ 5\mathrm{a}=0 \end{gathered}$$

Simplify further as follows.

$$\begin{gathered} \mathrm{a}=0 \\ \ker \left(\mathrm{T}\right)=\left\{\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\right\} \\ \ker \left(\mathrm{T}\right)=\left\{0\right\} \end{gathered}$$

Now, to check that if role="math" localid="1659852728461" $\mathfrak{J}\left(\mathrm{T}\right)={\mathrm{R}}^{2脳2}$as follows.

$\mathfrak{J}\left(\mathrm{T}\right)=\left\{\mathrm{T}\left(\mathrm{A}\right)|\mathrm{A}鈭�{\mathrm{R}}^{2脳2}\right\}$

Consider$\mathrm{A}鈭�{\mathrm{R}}^{2脳2}$be an arbitrary matrix as follows.

$\mathrm{T}\left(\mathrm{A}\right)=\left[\begin{array}{cc}2\mathrm{a}& 3\mathrm{a}\\ 4\mathrm{a}& 5\mathrm{a}\end{array}\right]$

Since, the image of the transformation is for every matrix from ${\mathrm{R}}^{2脳2}$there is no real number a and so T (A) = B holds

And the matric does not exist real number a.

$\mathfrak{J}\left(\mathrm{T}\right)鈮�{\mathrm{R}}^{2脳2}$

Therefore,T is not an isomorphism.

Thus, is a linear transformation and is not an isomorphism and$\ker \left(\mathrm{t}\right)=\left\{0\right\}$ also $\mathfrak{J}\left(\mathrm{T}\right)鈮�{\mathrm{R}}^{2脳2}$.