91Ó°ÊÓ

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

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

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

Consider the transformation as follows.

$\mathrm{T}:{\mathrm{R}}^{\mathrm{n}×\mathrm{m}}→\mathrm{F}({\mathrm{R}}^{\mathrm{n}},{\mathrm{R}}^{\mathrm{m}})$is defined by $\left(T\right(A\left)\right)(\stackrel{→}{v})=A\stackrel{→}{v}$.

Simplify for the linear transformation first condition as follows.

$$\begin{gathered} \left(\mathrm{T}\left(\mathrm{A}\right)\right)\left(\stackrel{→}{\mathrm{v}}+\stackrel{→}{\mathrm{w}}\right)=\mathrm{A}\left(\stackrel{→}{\mathrm{v}}+\stackrel{→}{\mathrm{w}}\right) \\ =\mathrm{A}\stackrel{→}{\mathrm{v}}+\mathrm{A}\stackrel{→}{\mathrm{w}} \\ \left(\mathrm{T}\left(\mathrm{A}\right)\right)\left(\stackrel{→}{\mathrm{v}}+\stackrel{→}{\mathrm{w}}\right)=\left(\mathrm{T}\left(\mathrm{A}\right)\right)\stackrel{→}{\mathrm{v}}+\left(\mathrm{T}\left(\mathrm{A}\right)\right)\stackrel{→}{\mathrm{w}} \end{gathered}$$

Similarly, simplify for the second condition as follows.

$$\begin{gathered} \left(T\left(A\right)\right)\left(k\stackrel{→}{v}\right)=A\left(k\stackrel{→}{v}\right) \\ =k\left(A\stackrel{→}{v}\right) \\ =K\left(T\left(A\right)\right)\left(\stackrel{→}{v}\right) \\ \left(T\left(A\right)\right)\left(k\stackrel{→}{v}\right)=k\left(T\left(A\right)\right)\left(\stackrel{→}{v}\right) \end{gathered}$$

Thus, T is a linear transformation.

A linear transformation$\mathbf{T}\mathbf{:}\mathbf{V}\mathbf{→}\mathbf{W}$ is said to be an isomorphism if and only if$\mathbf{ker}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathbf{0}\mathbf{\right\}}$ and$\mathbf{im}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{W}$ or $\mathbf{dim}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathbf{dim}\mathbf{\left(}\mathbf{W}\mathbf{\right)}$.

Consider for a non-zero vector$\stackrel{→}{v}$as follows.

$$\begin{gathered} \left(T\left(A\right)\right)\left(\stackrel{→}{v}\right)=\stackrel{→}{0} \\ A\stackrel{→}{v}=0 \end{gathered}$$

Then by equality of two matrix as follows.

$\stackrel{→}{v}=\stackrel{→}{0}$

Thus, the kernel of T contains only zero matrix.

A linear transformation$\mathbf{T}\mathbf{:}\mathbf{V}\mathbf{→}\mathbf{W}$is said to be an isomorphism if and only if$\mathbf{ker}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathbf{0}\mathbf{\right\}}$ and$\mathbf{im}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{W}$ or $\mathbf{dim}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathbf{dim}\mathbf{\left(}\mathbf{W}\mathbf{\right)}$.

Since, the transformation is linear.

Therefore, the image of T is the linear space of all the linear transformation from${\mathrm{R}}^{\mathrm{m}}$ to ${\mathrm{R}}^{\mathrm{n}}$.

Hence, the image of T is the space$\mathrm{L}({\mathrm{R}}^{\mathrm{m}},{\mathrm{R}}^{\mathrm{n}})$ of all linear transformation from${\mathrm{R}}^{\mathrm{m}}$ to ${\mathrm{R}}^{\mathrm{n}}$.

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

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

For all elements$\mathbf{f}\mathbf{,}\mathbf{g}$ of V and k is scalar.

A linear transformation$\mathbf{T}\mathbf{:}\mathbf{V}\mathbf{→}\mathbf{W}$ is said to be an isomorphism if and only if$\mathbf{ker}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathbf{0}\mathbf{\right\}}$ and$\mathbf{im}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{W}$ or $\mathbf{dim}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathbf{dim}\mathbf{\left(}\mathbf{W}\mathbf{\right)}$.

Consider the transformation as follows.

$\mathrm{T}:{\mathrm{R}}^{\mathrm{n}×\mathrm{m}}→\mathrm{F}\left({\mathrm{R}}^{\mathrm{n}},{\mathrm{R}}^{\mathrm{m}}\right)$is defined by $\left(\mathrm{T}\left(\mathrm{A}\right)\right)\left(\stackrel{→}{\mathrm{v}}\right)=\mathrm{A}\stackrel{→}{\mathrm{v}}$.

The dimension of the space is as follows.

$$\begin{gathered} \dim \left(\mathrm{L}\left({\mathrm{R}}^{\mathrm{m}},{\mathrm{R}}^{\mathrm{n}}\right)\right)=\dim \left({\mathrm{R}}^{\mathrm{n}×\mathrm{m}}\right) \\ =\mathrm{mn} \\ \dim \left(\mathrm{L}\left({\mathrm{R}}^{\mathrm{m}},{\mathrm{R}}^{\mathrm{n}}\right)\right)=\mathrm{mn} \end{gathered}$$

Thus, the dimension of$\mathrm{L}({\mathrm{R}}^{\mathrm{m}},{\mathrm{R}}^{\mathrm{n}})$ is mn.