Question

In Exercises 40 through 46, consider vectors$\overrightarrow{{\mathbf{V}}_{\mathbf{1}}}\mathbf{,}\overrightarrow{{\mathbf{V}}_{\mathbf{2}}}\mathbf{,}\overrightarrow{{\mathbf{V}}_{\mathbf{3}}}$in${\mathbf{R}}^{\mathbf{4}}$; we are told that$\overrightarrow{{\mathbf{V}}_{\mathbf{i}}}\mathbf{,}\overrightarrow{{\mathbf{V}}_{\mathbf{j}}}$is the entry ${\mathbf{a}}_{\mathbf{ij}}$of matrix A.

$\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find ${\mathbf{proj}}_{\overrightarrow{\mathbf{v}}}\left({\overrightarrow{v}}_1\right)$, expressed as a scalar multiple of${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}$.

Short answer

The required projection is$pro{j}_{{\overrightarrow{v}}_2}\left({\overrightarrow{v}}_1\right)=\frac{5}{9}{\overrightarrow{v}}_2$

Step-by-step solution

Formula for the orthogonal projection.

If Vis a subspace of ${\mathbf{R}}^{\mathbf{n}}$with an orthonormal basis ${\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{u}}}_{\mathbf{m}}$thenlocalid="1659451887578" ${\mathbf{proj}}_{\mathbf{v}}\overrightarrow{\mathbf{x}}\mathbf{=}{\overrightarrow{\mathbf{x}}}^{\mathbf{ll}}\mathbf{=}\mathbf{(}{\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{.}\overrightarrow{\mathbf{x}}\mathbf{)}{\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\mathbf{(}{\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{.}\overrightarrow{\mathbf{x}}\mathbf{)}{\overrightarrow{\mathbf{u}}}_{\mathbf{m}}$

For all $\overrightarrow{x}$in localid="1659436140715" $R^n$.

Let us write the matrix in the $\overrightarrow{V_i}.\overrightarrow{V_i}$notation.

Consider the terms below.

localid="1659451940364" $A=\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]=\left[\begin{array}{ccc}{\overrightarrow{v}}_1.{\overrightarrow{v}}_1& {\overrightarrow{v}}_1.{\overrightarrow{v}}_2& {\overrightarrow{v}}_1.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_2.{\overrightarrow{v}}_1& {\overrightarrow{v}}_2.{\overrightarrow{v}}_2& {\overrightarrow{v}}_2.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_3.{\overrightarrow{v}}_1& {\overrightarrow{v}}_3.{\overrightarrow{v}}_2& {\overrightarrow{v}}_3.{\overrightarrow{v}}_3\end{array}\right]=\left[\begin{array}{ccc}{||{\overrightarrow{v}}_1||}^2& {\overrightarrow{v}}_1.{\overrightarrow{v}}_2& {\overrightarrow{v}}_1.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_2.{\overrightarrow{v}}_1& {||{\overrightarrow{v}}_2||}^2& {\overrightarrow{v}}_2.{\overrightarrow{v}}_3\\ {\overrightarrow{v}}_3.{\overrightarrow{v}}_1& {\overrightarrow{v}}_3.{\overrightarrow{v}}_2& {||{\overrightarrow{v}}_3||}^2\end{array}\right]$

obtain the orthonormal basis for span(v→2)

To obtain the span. Since it is a singleton vector. So by normalizing it. Therefore, an orthonormal basis will be.

$\frac{{\overrightarrow{v}}_2}{||{\overrightarrow{v}}_2||}=\frac{{\overrightarrow{v}}_2}{3}=\overrightarrow{u}$.

So, according to the above theorem

$$\begin{gathered} pro{j}_{{\overrightarrow{v}}_2}\left({\overrightarrow{v}}_1\right)=\left(\overrightarrow{u}.{\overrightarrow{v}}_1\right)\overrightarrow{u} \\ =\frac{{\overrightarrow{v}}_2}{3}.{\overrightarrow{v}}_1\frac{{\overrightarrow{v}}_2}{3} \\ =\frac{{\overrightarrow{v}}_2.{\overrightarrow{v}}_1}{3}\frac{{\overrightarrow{v}}_2}{3} \\ =\frac{5}{9}{\overrightarrow{v}}_2 \end{gathered}$$

Hence, the role="math" localid="1659438206334" $pro{j}_{{\overrightarrow{v}}_2}\left({\overrightarrow{v}}_1\right)=\frac{5}{9}{\overrightarrow{v}}_2$.