91Ó°ÊÓ

Consider two linear spaces Vand W. A function Tfrom Vto Wis called linear transformation if

1. $\mathbf{T}\mathbf{(}\mathbf{f}\mathbf{+}\mathbf{g}\mathbf{)}\mathbf{=}\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{+}\mathbf{T}\mathbf{\left(}\mathbf{g}\mathbf{\right)}$
2. $\mathbf{T}\mathbf{\left(}\mathbf{kf}\mathbf{\right)}\mathbf{=}\mathbf{kT}\mathbf{\left(}\mathbf{f}\mathbf{\right)}$

for all elements f and g of Vand for all scalars.

If Vis finite dimensional, then

$\mathbf{dim}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathbf{rank}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{+}\mathbf{nullity}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{dim}\mathbf{\left(}\mathbf{imT}\mathbf{\right)}\mathbf{+}\mathbf{dim}\mathbf{\left(}\mathbf{kerT}\mathbf{\right)}$

A linear transformation $\mathbf{T}\mathbf{:}\mathbf{V}\mathbf{→}\mathbf{W}$is called isomorphism if its both one-to-one and onto.

Let$x,y∈V$ and assume $T\left(x\right)=T\left(y\right)$.

Then,

$$\begin{gathered} x=l_v\left(x\right) \\ =\left(T^{-1}∘T\right)\left(x\right)=T^{-1}\left(T\left(y\right)\right) \\ =\left(T^{-1}∘T\right)\left(y\right)=l_{v}y=y \end{gathered}$$

Therefore, T is one-to-one.

Suppose,$v∈W$ then$T^{-1}\left(v\right)$ is a vector in $T^{-1}\left(v\right)$.

$$\begin{gathered} T\left(T^{-1}\left(v\right)\right)=\left(T∘T^{-1}\right)\left(v\right) \\ =l_w\left(v\right)=v \end{gathered}$$

Therefore,T is onto.

Since T is both one-to-one and onto, T is an isomorphism from V to W.

Let T be an isomorphism.

Consider two scalars a,b then,

$T^{-1}\left(aT\left({\stackrel{→}{v}}_i\right)+bT\left({\stackrel{→}{v}}_2\right)\right)$

Where localid="1659428486388" $\stackrel{→}{v}{l}_1.{\stackrel{→}{v}}_2∈V$.

localid="1659428490353" $$\begin{gathered} T^{-1}\left(aT\left({\stackrel{→}{v}}_1\right)+bT\left({\stackrel{→}{v}}_2\right)\right)=a{T}^{-1}\left(T\left({\stackrel{→}{v}}_1\right)\right)+b{T}^{-1}\left(T\left({\stackrel{→}{v}}_2\right)\right) \\ =a{\stackrel{→}{v}}_1+b{\stackrel{→}{v}}_2 \end{gathered}$$

Since T is one-to-one,

localid="1659428481689" $$\begin{gathered} T\left(a{\stackrel{→}{v}}_1+b{\stackrel{→}{v}}_2\right)=T\left(T^{-1}\left(aT\left({\stackrel{→}{v}}_1\right)+bT\left({\stackrel{→}{v}}_2\right)\right)\right) \\ =aT\left({\stackrel{→}{v}}_1\right)+bT\left({\stackrel{→}{v}}_2\right) \end{gathered}$$

Since, T is a linear map, is also a linear map.

Iflocalid="1659428501365" $\stackrel{→}{v}∈V$is given, then localid="1659428494771" $\stackrel{→}{v}=T^{-1}\left(T\left(\stackrel{→}{v}\right)\right)$.

Therefore,localid="1659428505699" $T{}^{-1}$is onto.

Iflocalid="1659428511347" $T^{-1}\left(\stackrel{→}{v}\right)=0$then,

localid="1659428515764" $\stackrel{→}{v}=T\left(T^{-1}\left(\stackrel{→}{v}\right)\right)=T\left(\stackrel{→}{0}\right)=\left(\stackrel{→}{0}\right)$

Thus,localid="1659428524692" $T^{-1}$is ono-to-one.

Therefore, localid="1659428520710" $T{}^{-1}$is an isomorphism from W to V.