Short Answer

In Exercises 40 through 46, consider vectors $\vec{V_{1}}, \vec{V_{2}}, \vec{V_{3}}$ in $R^{4}$ ; we are told that $\vec{V_{i}}, \vec{V_{j}}$ is the entry $a_{ij}$ of matrix A.
$A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$

Find ${proj}_{\vec{v}} ({\vec{v}}_{1})$ , expressed as a scalar multiple of ${\vec{v}}_{2}$ .

The required projection is $p r o j_{{\vec{v}}_{2}} ({\vec{v}}_{1}) = \frac{5}{9} {\vec{v}}_{2}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Formula for the orthogonal projection.

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

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

Let us write the matrix in the $\vec{V_{i}} . \vec{V_{i}}$ notation.

Consider the terms below.

localid="1659451940364" $A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}] = [\begin{matrix} {\vec{v}}_{1} . {\vec{v}}_{1} & {\vec{v}}_{1} . {\vec{v}}_{2} & {\vec{v}}_{1} . {\vec{v}}_{3} \\ {\vec{v}}_{2} . {\vec{v}}_{1} & {\vec{v}}_{2} . {\vec{v}}_{2} & {\vec{v}}_{2} . {\vec{v}}_{3} \\ {\vec{v}}_{3} . {\vec{v}}_{1} & {\vec{v}}_{3} . {\vec{v}}_{2} & {\vec{v}}_{3} . {\vec{v}}_{3} \end{matrix}] = [\begin{matrix} {||{\vec{v}}_{1}||}^{2} & {\vec{v}}_{1} . {\vec{v}}_{2} & {\vec{v}}_{1} . {\vec{v}}_{3} \\ {\vec{v}}_{2} . {\vec{v}}_{1} & {||{\vec{v}}_{2}||}^{2} & {\vec{v}}_{2} . {\vec{v}}_{3} \\ {\vec{v}}_{3} . {\vec{v}}_{1} & {\vec{v}}_{3} . {\vec{v}}_{2} & {||{\vec{v}}_{3}||}^{2} \end{matrix}]$

Step 2: 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.

$\frac{{\vec{v}}_{2}}{||{\vec{v}}_{2}||} = \frac{{\vec{v}}_{2}}{3} = \vec{u}$ .

So, according to the above theorem

$p r o j_{{\vec{v}}_{2}} ({\vec{v}}_{1}) = (\vec{u} . {\vec{v}}_{1}) \vec{u} = \frac{{\vec{v}}_{2}}{3} . {\vec{v}}_{1} \frac{{\vec{v}}_{2}}{3} = \frac{{\vec{v}}_{2} . {\vec{v}}_{1}}{3} \frac{{\vec{v}}_{2}}{3} = \frac{5}{9} {\vec{v}}_{2}$

Hence, the role="math" localid="1659438206334" $p r o j_{{\vec{v}}_{2}} ({\vec{v}}_{1}) = \frac{5}{9} {\vec{v}}_{2}$ .

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Linear Algebra With Applications

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Step by Step Solution

Step 1: Formula for the orthogonal projection.

Step 2: obtain the orthonormal basis for span(v→2)

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Linear Algebra With Applications

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Short Answer

In Exercises 40 through 46, consider vectorsV1→,V2→,V3→ inR4 ; we are told thatVi→,Vj→ is the entry aijof matrix A.A=[35115920112049] Find projv→(v→1), expressed as a scalar multiple ofv→2 .

Step by Step Solution

Step 1: Formula for the orthogonal projection.

Step 2: obtain the orthonormal basis for span(v→2)

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