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If Vis a subspace of${\mathit{R}}^{\mathbf{n}}$ with an orthonormal basisrole="math" localid="1659496644664" ${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}$ then

role="math" localid="1659496850706" $\mathit{p}\mathit{r}\mathit{o}{\mathbf{j}}_{\mathbf{v}}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}{\stackrel{\mathbf{鈫�}}{\mathbf{x}}}^{\mathbf{\left|}\mathbf{\right|}}\mathbf{=}\left({\stackrel{鈫�}{u}}_1.\stackrel{鈫�}{x}\right){\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\left({\stackrel{鈫�}{u}}_m.\stackrel{鈫�}{x}\right){\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}$

For all$\stackrel{\mathbf{鈫�}}{\mathbf{x}}$in${\mathit{R}}^{\mathbf{n}}$.

Let us write the matrix in the${\stackrel{鈫�}{v}}_i.{\stackrel{鈫�}{v}}_j$ notation.

Consider the terms below.

$A=\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]=\left[\begin{array}{ccc}{\stackrel{鈫�}{v}}_1.{\stackrel{鈫�}{v}}_1& {\stackrel{鈫�}{v}}_1.{\stackrel{鈫�}{v}}_2& {\stackrel{鈫�}{v}}_1.{\stackrel{鈫�}{v}}_3\\ {\stackrel{鈫�}{v}}_2.{\stackrel{鈫�}{v}}_1& {\stackrel{鈫�}{v}}_2.{\stackrel{鈫�}{v}}_2& {\stackrel{鈫�}{v}}_2.{\stackrel{鈫�}{v}}_3\\ {\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_1& {\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_2& {\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_3\end{array}\right]=\left[\begin{array}{ccc}{\left|\left|{\stackrel{鈫�}{v}}_1\right|\right|}^2& {\stackrel{鈫�}{v}}_1.{\stackrel{鈫�}{v}}_2& {\stackrel{鈫�}{v}}_1.{\stackrel{鈫�}{v}}_3\\ {\stackrel{鈫�}{v}}_2.{\stackrel{鈫�}{v}}_1& {\left|\left|{\stackrel{鈫�}{v}}_2\right|\right|}^2& {\stackrel{鈫�}{v}}_2.{\stackrel{鈫�}{v}}_3\\ {\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_1& {\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_2& {\left|\left|{\stackrel{鈫�}{v}}_3\right|\right|}^2\end{array}\right]$

Since, here is need to express the required projection as a linear combination of${\stackrel{鈫�}{v}}_1$ and${\stackrel{鈫�}{v}}_2$, so let us assume that for some $伪,尾鈭�\mathrm{鈩�}$. Then, $pro{j}_v\left({\stackrel{鈫�}{v}}_3\right)=伪{\stackrel{鈫�}{v}}_1+尾{\stackrel{鈫�}{v}}_2$.

In the above term. Here find out the value of $伪,尾$.

Consider the equations below to find the value of$伪$ and $尾$.

$$\begin{gathered} \left({\stackrel{鈫�}{v}}_{3}pro{j}_v\left({\stackrel{鈫�}{v}}_3\right)\right)鈯�V鈬�\left({\stackrel{鈫�}{v}}_3-伪{\stackrel{鈫�}{v}}_1-尾{\stackrel{鈫�}{v}}_2\right).{\stackrel{鈫�}{v}}_1=0 \\ 鈬�{\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_1-伪{\left|\left|{\stackrel{鈫�}{v}}_2\right|\right|}^2-尾{\stackrel{鈫�}{v}}_2.{\stackrel{鈫�}{v}}_1=0 \\ 鈬�11-3伪-5尾=0 \\ 鈬�3伪+5尾=11......\left(1\right) \end{gathered}$$

Similarly,

$$\begin{gathered} \left({\stackrel{鈫�}{v}}_{3}pro{j}_v\left({\stackrel{鈫�}{v}}_3\right)\right)鈯�V鈬�\left({\stackrel{鈫�}{v}}_3-伪{\stackrel{鈫�}{v}}_1-尾{\stackrel{鈫�}{v}}_2\right).{\stackrel{鈫�}{v}}_2=0 \\ 鈬�{\stackrel{鈫�}{v}}_3.{\stackrel{鈫�}{v}}_2-伪{\stackrel{鈫�}{v}}_1.{\stackrel{鈫�}{v}}_2-尾{\left|\left|{\stackrel{鈫�}{v}}_2\right|\right|}^2=0 \\ 鈬�20-5伪-9尾=0 \\ 鈬�5伪+9尾=20......\left(2\right) \end{gathered}$$

Now, consider the equation (1) and (2) by solving them find out the value of .

$伪=\frac{-1}{2}$and$尾=\frac{5}{2}$

Hence, the required projection is $pro{j}_v\left({\stackrel{鈫�}{v}}_3\right)=\frac{-1}{2}{\stackrel{鈫�}{v}}_1-\frac{5}{2}{\stackrel{鈫�}{v}}_2$.