91影视

• A linear transformation is a function that goes from one vector space to another while keeping the underlying (linear) structure of each vector space in mind.
• A linear transformation can also be referred to as a linear operator or map.

Write $L\left(\stackrel{鈬赌}{x}\right)=A\stackrel{鈬赌}{x}$for all $\stackrel{鈬赌}{x}鈭�{\mathrm{鈩�}}^m$, where A is the matrix of transformation L in some basis, now, we observe that $A^{T}A$is symmetric, since

${\left(A^{T}A\right)}^T=A^T{\left(A^T\right)}^T$

Note: The transpose of the transpose of a matrix is the matrix itself.

role="math" localid="1660732747614" $=A^{T}A$

Since $A^{T}A$is a symmetric matrix, then there exists an orthonormal eigenbasis ${\stackrel{鈬赌}{v}}_1,...,{\stackrel{鈬赌}{v}}_m$of ${\mathrm{鈩�}}^m$for it.

That is, we have $A^{T}A{\stackrel{鈬赌}{v}}_i={位}_i{\stackrel{鈬赌}{v}}_i$.

Now consider the vectors $A{\stackrel{鈬赌}{v}}_1,A{\stackrel{鈬赌}{v}}_2,...,A{\stackrel{鈬赌}{v}}_m$, these vectors are orthogonal since

$$\begin{gathered} \left(A{\stackrel{鈬赌}{v}}_i\right).\left(A{\stackrel{鈬赌}{v}}_j\right)={\left(A{\stackrel{鈬赌}{v}}_i\right)}^T.\left(A{\stackrel{鈬赌}{v}}_j\right) \\ ={\left({\stackrel{鈬赌}{v}}_i\right)}^T\left(A^{T}A{\stackrel{鈬赌}{v}}_j\right) \\ ={\left({\stackrel{鈬赌}{v}}_i\right)}^T\left({位}_j{\stackrel{鈬赌}{v}}_j\right) \\ ={位}_j\left[{\left({\stackrel{鈬赌}{v}}_i\right)}^T\left({\stackrel{鈬赌}{v}}_j\right)\right] \\ \mathrm{Note}:\mathrm{If} \\ \stackrel{鈬赌}{\mathrm{v}}=\left[\begin{array}{c}{\mathrm{v}}_1\\ 鈰�\\ {\mathrm{v}}_{\mathrm{n}}\end{array}\right] \\ \mathrm{Then} \\ {\stackrel{鈬赌}{\mathrm{v}}}^{\mathrm{T}}\stackrel{鈬赌}{\mathrm{w}}=\left[{\mathrm{v}}_1...{\mathrm{v}}_{\mathrm{n}}\right]\left[\begin{array}{c}{\mathrm{w}}_1\\ 鈰�\\ {\mathrm{w}}_{\mathrm{n}}\end{array}\right]={\stackrel{鈬赌}{\mathrm{v}}}_1{\stackrel{鈬赌}{\mathrm{w}}}_1+...+{\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{n}}{\stackrel{鈬赌}{\mathrm{w}}}_{\mathrm{n}}=\stackrel{鈬赌}{\mathrm{v}}.\stackrel{鈬赌}{\mathrm{w}} \\ \mathrm{Then}\mathrm{in}\mathrm{our}\mathrm{case}{\left({\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{i}}\right)}^{\mathrm{T}}.\left({\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{j}}\right)={\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{i}}.{\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{j}} \\ ={\mathrm{位}}_{\mathrm{j}}\left[{\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{i}}.{\stackrel{鈬赌}{\mathrm{v}}}_{\mathrm{j}}\right] \\ =0 \\ \mathrm{For}\mathrm{all}i鈮�j, \\ \mathrm{Write}\mathrm{L}\left(\stackrel{鈬赌}{\mathrm{x}}\right)=\mathrm{A}\stackrel{鈬赌}{\mathrm{x}}\mathrm{for}\mathrm{some}\mathrm{matrix}A \end{gathered}$$