91Ó°ÊÓ

If Vis a subspace of ${\mathit{R}}^{\mathbf{n}}$with an orthonormal basis $\mathit{u}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{u}{\mathbf{⃗}}_{\mathbf{m}}$then

$\mathit{p}\mathit{r}\mathit{o}{\mathit{j}}_{\mathbf{v}}\mathit{x}\mathbf{⃗}\mathbf{=}{\mathbf{x}\mathbf{⃗}\mathbf{}}^{\mathbf{II}}\mathbf{=}\mathbf{(}\mathit{u}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{.}\mathit{x}\mathbf{⃗}\mathbf{)}\mathit{u}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\mathbf{(}\mathit{u}{\mathbf{⃗}}_{\mathbf{m}}\mathbf{.}\mathit{x}\mathbf{⃗}\mathbf{)}\mathit{u}{\mathbf{⃗}}_{\mathbf{m}}$

For all $\stackrel{→}{x}⃗$in $R^n.$

Let us write the matrix in the$v{⃗}_i.v{⃗}_j$notation.

Consider the terms below.

$$\begin{gathered} A=\left[3511 \\ 5920 \\ 112049\right]=\left[{\stackrel{â‡\P Ä}{V}}_1.{\stackrel{â‡\P Ä}{V}}_1{\stackrel{â‡\P Ä}{V}}_1.{\stackrel{â‡\P Ä}{V}}_2{\stackrel{â‡\P Ä}{V}}_1.{\stackrel{â‡\P Ä}{V}}_3 \\ {\stackrel{â‡\P Ä}{V}}_2.{\stackrel{â‡\P Ä}{V}}_1{\stackrel{â‡\P Ä}{V}}_2.{\stackrel{â‡\P Ä}{V}}_2{\stackrel{â‡\P Ä}{V}}_2.{\stackrel{â‡\P Ä}{V}}_3 \\ {\stackrel{â‡\P Ä}{V}}_3.{\stackrel{â‡\P Ä}{V}}_1{\stackrel{â‡\P Ä}{V}}_3.{\stackrel{â‡\P Ä}{V}}_2{\stackrel{â‡\P Ä}{V}}_3.{\stackrel{â‡\P Ä}{V}}_3\right]=\left[{\left|\left|{\stackrel{â‡\P Ä}{V}}_1\right|\right|}^2{\stackrel{â‡\P Ä}{V}}_1.{\stackrel{â‡\P Ä}{V}}_2{\stackrel{â‡\P Ä}{V}}_1.{\stackrel{â‡\P Ä}{V}}_3 \\ {\stackrel{â‡\P Ä}{V}}_2.{\stackrel{â‡\P Ä}{V}}_1\left|\left|{\stackrel{â‡\P Ä}{V}}_2\right|\right|{\stackrel{â‡\P Ä}{V}}_2.{\stackrel{â‡\P Ä}{V}}_3 \\ {\stackrel{â‡\P Ä}{V}}_3.{\stackrel{â‡\P Ä}{V}}_1{\stackrel{â‡\P Ä}{V}}_3.{\stackrel{â‡\P Ä}{V}}_2{\left|\left|{\stackrel{â‡\P Ä}{V}}_3\right|\right|}^2\right] \end{gathered}$$

Since, here is need to express the required projection as a linear combination$v{⃗}_2$of and , $v{⃗}_3$so let us assume that for some$\mathrm{Î\pm },\mathrm{β}∈R.$. Then,$pro{j}_v\left(v{⃗}_1\right)=\mathrm{Î\pm }v{⃗}_2+\mathrm{β}v{⃗}_3.$.

In the above term. Here find out the value of $\mathrm{Î\pm },\mathrm{β}.$.

Consider the equations below to find the value of and $\mathrm{β}$.

$$\begin{gathered} \left({\stackrel{â‡\P Ä}{v}}_{1}pro{j}_{\mathrm{v}}\left({\stackrel{â‡\P Ä}{v}}_1\right)\right)âŠ\yen V⇒\left({\stackrel{â‡\P Ä}{V}}_1-a{\stackrel{â‡\P Ä}{v}}_2-\mathrm{β}{\stackrel{â‡\P Ä}{v}}_3\right).{\stackrel{â‡\P Ä}{v}}_2=0 \\ ⇒{\stackrel{â‡\P Ä}{v}}_1.{\stackrel{â‡\P Ä}{v}}_2-a{\left|\left|{\stackrel{â‡\P Ä}{v}}_2\right|\right|}^2-\mathrm{β}{\stackrel{â‡\P Ä}{v}}_3.{\stackrel{â‡\P Ä}{v}}_2=0 \\ ⇒5-9a-20\mathrm{β}=0...........\left(2\right) \end{gathered}$$

Similarly,

$$\begin{gathered} \left({\stackrel{â‡\P Ä}{v}}_{1}pro{j}_{\mathrm{v}}\left({\stackrel{â‡\P Ä}{v}}_1\right)\right)âŠ\yen V⇒\left({\stackrel{â‡\P Ä}{V}}_1-a{\stackrel{â‡\P Ä}{v}}_2-\mathrm{β}{\stackrel{â‡\P Ä}{v}}_3\right).{\stackrel{â‡\P Ä}{v}}_2=0 \\ ⇒{\stackrel{â‡\P Ä}{v}}_1.{\stackrel{â‡\P Ä}{v}}_3-a{\stackrel{â‡\P Ä}{v}}_2.{\stackrel{â‡\P Ä}{v}}_3-\mathrm{β}{\left|\left|{\stackrel{â‡\P Ä}{v}}_3\right|\right|}^2=0 \\ ⇒11-20a-49\mathrm{β}=11 \\ ⇒20a+49\mathrm{β}=11......\left(1\right) \end{gathered}$$

Now, consider the equation (1) and (2) by solving them find out the $a,\mathrm{β}$value of

$a=\frac{25}{41}$and $\mathrm{β}=\frac{-1}{41}$.

Thus, the required projection is $pro{j}_v\left({\stackrel{â‡\P Ä}{v}}_1\right)=\frac{25}{41}{\stackrel{â‡\P Ä}{v}}_2-\frac{1}{41}{\stackrel{â‡\P Ä}{v}}_3$.