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

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

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

An invertible linear transformation is called an isomorphism.

The given transformation as follows.

T (M) = sum of diagonal entries from${\mathrm{R}}^{2\mathrm{X}2}\mathrm{to}\mathrm{R}$

By using the definition of linear transformation as follows.

$\mathrm{T}\left(\mathrm{A}+\mathrm{B}\right)=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right)$

Consider the 2X2 matrix for A and B as follows.

$$\begin{gathered} \mathrm{A}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right] \\ \mathrm{B}=\left[\begin{array}{cc}\mathrm{e}& \mathrm{f}\\ \mathrm{g}& \mathrm{h}\end{array}\right] \end{gathered}$$

Substitute the matrix $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$for A and $\left[\begin{array}{cc}\mathrm{e}& \mathrm{f}\\ \mathrm{g}& \mathrm{h}\end{array}\right]$for B in$\mathrm{T}\left(\mathrm{A}+\mathrm{B}\right)=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right)$ as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{A}+\mathrm{B}\right)=\mathrm{A}+\mathrm{B} \\ =\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]+\left[\begin{array}{cc}\mathrm{e}& \mathrm{f}\\ \mathrm{g}& \mathrm{h}\end{array}\right] \\ =\left[\begin{array}{cc}\mathrm{a}+\mathrm{e}& \mathrm{b}+\mathrm{f}\\ \mathrm{c}+\mathrm{g}& \mathrm{d}+\mathrm{h}\end{array}\right] \\ =\mathrm{a}+\mathrm{d}+\mathrm{e}+\mathrm{h} \\ \mathrm{T}\left(\mathrm{A}+\mathrm{B}\right)=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right) \end{gathered}$$

Now, simplify for scalars as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{kA}\right)=\mathrm{T}\left[\begin{array}{cc}\mathrm{ka}& \mathrm{kb}\\ \mathrm{kc}& \mathrm{kd}\end{array}\right] \\ =\mathrm{ka}+\mathrm{kd} \\ =\mathrm{k}\left(\mathrm{a}+\mathrm{d}\right) \\ \mathrm{T}\left(\mathrm{kA}\right)=\mathrm{kT}\left(\mathrm{A}\right) \end{gathered}$$

This shows that T the transformation.

The linear space V is isomorphism if dim (V) =dim (w) and only if .

The property says that T cannot be isomorphism between ${\mathrm{R}}^{2\mathrm{X}2}\mathrm{to}\mathrm{R}$

Because the dimension of first space is 4 and the dimension of second space is 1.

Hence, the transformation T is linear and is not isomorphism.