91影视

Consider the function T defined as$T=\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ 0& a_{22}& a_{23}& a_{24}\\ 0& 0& a_{33}& a_{34}\\ 0& 0& 0& a_{44}\end{array}\right]\left[\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{12}& a_{22}& 0& 0\\ a_{13}& a_{23}& a_{33}& 0\\ a_{14}& a_{24}& a_{34}& a_{44}\end{array}\right]$

A function Dis called a linear transformation on Rif the function Dsatisfies the following property鈥檚.

1. $\mathbf{D}\left(\mathbf{x}\mathbf{+}\mathbf{y}\right)\mathbf{=}\mathbf{D}\left(\mathbf{x}\right)\mathbf{+}\mathbf{D}\left(\mathbf{y}\right)$for all$\mathbf{x}\mathbf{,}\mathbf{y}鈭�\mathrm{鈩�}$.
2. $\mathbf{D}\left(\mathbf{伪虫}\right)\mathbf{=}\mathbf{伪顿}\left(\mathbf{x}\right)$for all constent$\mathbf{伪}鈭�\mathrm{鈩�}$.

An invertible linear transformation is called isomorphism or dimension of domain and co-domain is not same then the function is not isomorphism

Simplify$T\left[\left[\begin{array}{cccc}{\left(a_{11}\right)}_1+{\left(a_{11}\right)}_2& {\left(a_{12}\right)}_1+{\left(a_{12}\right)}_2& {\left(a_{13}\right)}_1+{\left(a_{13}\right)}_2& {\left(a_{14}\right)}_1+{\left(a_{14}\right)}_2\\ 0& {\left(a_{12}\right)}_1+{\left(a_{12}\right)}_2& {\left(a_{23}\right)}_1+{\left(a_{23}\right)}_2& {\left(a_{24}\right)}_1+{\left(a_{24}\right)}_2\\ 0& 0& {\left(a_{33}\right)}_1+{\left(a_{33}\right)}_2& {\left(a_{34}\right)}_1+{\left(a_{34}\right)}_2\\ 0& 0& 0& {\left(a_{44}\right)}_1+{\left(a_{44}\right)}_2\end{array}\right]\right]$ as follows.

$$\begin{gathered} T\left[\left[\begin{array}{cccc}{\left(a_{11}\right)}_1+{\left(a_{11}\right)}_2& {\left(a_{12}\right)}_1+{\left(a_{12}\right)}_2& {\left(a_{13}\right)}_1+{\left(a_{13}\right)}_2& {\left(a_{14}\right)}_1+{\left(a_{14}\right)}_2\\ 0& {\left(a_{12}\right)}_1+{\left(a_{12}\right)}_2& {\left(a_{23}\right)}_1+{\left(a_{23}\right)}_2& {\left(a_{24}\right)}_1+{\left(a_{24}\right)}_2\\ 0& 0& {\left(a_{33}\right)}_1+{\left(a_{33}\right)}_2& {\left(a_{34}\right)}_1+{\left(a_{34}\right)}_2\\ 0& 0& 0& {\left(a_{44}\right)}_1+{\left(a_{44}\right)}_2\end{array}\right]\right] \\ =\left[\begin{array}{cccc}{\left(a_{11}\right)}_1+{\left(a_{11}\right)}_2& 0& 0& 0\\ {\left(a_{12}\right)}_1+{\left(a_{12}\right)}_2& {\left(a_{22}\right)}_1+{\left(a_{22}\right)}_2& 0& 0\\ {\left(a_{13}\right)}_1+{\left(a_{13}\right)}_2& {\left(a_{23}\right)}_1+{\left(a_{23}\right)}_2& {\left(a_{33}\right)}_1+{\left(a_{33}\right)}_2& 0\\ {\left(a_{14}\right)}_1+{\left(a_{14}\right)}_2& {\left(a_{24}\right)}_1+{\left(a_{24}\right)}_2& {\left(a_{34}\right)}_1+{\left(a_{34}\right)}_2& {\left(a_{44}\right)}_1+{\left(a_{44}\right)}_2\end{array}\right]1 \\ =\left\{\left[\begin{array}{cccc}{\left(a_{11}\right)}_1& 0& 0& 0\\ {\left(a_{12}\right)}_1& {\left(a_{22}\right)}_1& 0& 0\\ {\left(a_{13}\right)}_1& {\left(a_{23}\right)}_1& {\left(a_{33}\right)}_1& 0\\ {\left(a_{14}\right)}_1& {\left(a_{24}\right)}_1& {\left(a_{34}\right)}_1& {\left(a_{44}\right)}_1\end{array}\right]+\left[\begin{array}{cccc}{\left(a_{11}\right)}_2& 0& 0& 0\\ {\left(a_{12}\right)}_2& {\left(a_{22}\right)}_2& 0& 0\\ {\left(a_{13}\right)}_2& {\left(a_{23}\right)}_2& {\left(a_{33}\right)}_2& 0\\ {\left(a_{14}\right)}_2& {\left(a_{24}\right)}_2& {\left(a_{34}\right)}_2& {\left(a_{44}\right)}_2\end{array}\right]\right\} \\ T=\left\{T\left[\begin{array}{cccc}{\left(a_{11}\right)}_1& {\left(a_{12}\right)}_1& {\left(a_{13}\right)}_1& {\left(a_{14}\right)}_1\\ & {\left(a_{22}\right)}_1& {\left(a_{23}\right)}_1& {\left(a_{24}\right)}_1\\ & & {\left(a_{33}\right)}_1& {\left(a_{34}\right)}_1\\ & & & {\left(a_{44}\right)}_1\end{array}\right]+T\left[\begin{array}{cccc}{\left(a_{11}\right)}_2& {\left(a_{12}\right)}_2& {\left(a_{13}\right)}_2& {\left(a_{14}\right)}_2\\ 0& {\left(a_{22}\right)}_2& {\left(a_{23}\right)}_2& {\left(a_{24}\right)}_2\\ 0& 0& {\left(a_{33}\right)}_2& {\left(a_{34}\right)}_2\\ 0& 0& 0& {\left(a_{44}\right)}_2\end{array}\right]\right\} \end{gathered}$$

Assume $\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ 0& a_{22}& a_{23}& a_{24}\\ 0& 0& a_{33}& a_{34}\\ 0& 0& 0& a_{44}\end{array}\right]鈭�R^{4脳4}$and $伪鈭�\mathrm{鈩�}$then .

$T\left(\left[\begin{array}{cccc}伪a_{11}& 伪a_{12}& 伪a_{13}& 伪a_{14}\\ 0& 伪a_{22}& 伪a_{23}& 伪a_{24}\\ 0& 0& 伪a_{33}& 伪a_{34}\\ 0& 0& 0& 伪a_{44}\end{array}\right]\right)=\left[\begin{array}{cccc}伪a_{11}& 0& 0& 0\\ 伪a_{12}& 伪a_{22}& 0& 0\\ 伪a_{13}& 伪a_{23}& 伪a_{33}& 0\\ 伪a_{14}& 伪a_{24}& 伪a_{34}& 伪a_{44}\end{array}\right]$

Simplify the equationrole="math" localid="1660407958523" $T\left(\left[\begin{array}{cccc}伪a_{11}& 伪a_{12}& 伪a_{13}& 伪a_{14}\\ 0& 伪a_{22}& 伪a_{23}& 伪a_{24}\\ 0& 0& 伪a_{33}& 伪a_{34}\\ 0& 0& 0& 伪a_{44}\end{array}\right]\right)=\left[\begin{array}{cccc}伪a_{11}& 0& 0& 0\\ 伪a_{12}& 伪a_{22}& 0& 0\\ 伪a_{13}& 伪a_{23}& 伪a_{33}& 0\\ 伪a_{14}& 伪a_{24}& 伪a_{34}& 伪a_{44}\end{array}\right]$as follows.

$\begin{array}{rcl}\left(\left[\begin{array}{cccc}伪a_{11}& 伪a_{12}& 伪a_{13}& 伪a_{14}\\ 0& 伪a_{22}& 伪a_{23}& 伪a_{24}\\ 0& 0& 伪a_{33}& 伪a_{34}\\ 0& 0& 0& 伪a_{44}\end{array}\right]\right)& =& \left[\begin{array}{cccc}伪a_{11}& 0& 0& 0\\ 伪a_{12}& 伪a_{22}& 0& 0\\ 伪a_{13}& 伪a_{23}& 伪a_{33}& 0\\ 伪a_{14}& 伪a_{24}& 伪a_{34}& 伪a_{44}\end{array}\right]\\ & =& 伪\left\{\left[\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{12}& a_{22}& 0& 0\\ a_{13}& a_{23}& a_{33}& 0\\ a_{14}& a_{24}& a_{34}& a_{44}\end{array}\right]\right\}\\ & =& 伪\left\{\left(\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ 0& a_{22}& a_{23}& a_{24}\\ 0& 0& a_{33}& a_{34}\\ 0& 0& 0& a_{44}\end{array}\right]\right)\right\}\end{array}$

As $T\left(M+N\right)=T\left(M\right)+T\left(N\right)$and$T\left(伪M\right)=伪\left\{T\left(M\right)\right\}$ , by the definition of linear transformation T is linear.

Theorem: Consider a linear transformation Tdefined from $\mathbf{T}\mathbf{:}\mathbf{V}鈫�\mathbf{W}$then the transformationT is an isomorphism if and only if $\mathbf{Ker}\left(\mathbf{T}\right)\mathbf{=}\mathbf{0}$where.

$\mathbf{Ker}\left(\mathbf{T}\right)\mathbf{=}\left\{\mathbf{f}\left(\mathbf{x}\right)鈭�\mathbf{P}\mathbf{:}\mathbf{T}\left\{\mathbf{f}\left(\mathbf{x}\right)\right\}\mathbf{=}\mathbf{0}\right\}$

As the dimension of $\text{kernel}\left(T\right)$is 0, by the theorem the function T is an isomorphism.

Hence, the statement is true.