Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

TRUE OR FALSE?
If a matrix $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ represents the orthogonal projection onto a line L , then the equation $a^{2} + b^{2} + c^{2} + d^{2} = 1$ must hold.

The given statement is false.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the parameters

The orthogonal projection of matrix is of form $(\begin{matrix} u_{1}^{2} & u_{1} u_{2} \\ u_{1} u_{2} & u_{2}^{2} \end{matrix})$ with $\vec{u} = [\begin{matrix} u_{1} \\ u_{2} \end{matrix}]$ . Thus, if

$a^{2} + b^{2} + c^{2} + d^{2} = 1 then$

${\begin{matrix} u_{1}^{4} + & (u_{1} u_{2}) \end{matrix}}^{2} + {(u_{1} u_{2})}^{2} + u_{2}^{4} = 1 \Rightarrow {(u_{1} u_{2})}^{2} = 1$

Consider the matrix.

$A = (\begin{matrix} a & b \\ c & d \end{matrix})$

Step 2: Find the values of the elements

If $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ then,

The elements will be,

$a = u_{1}^{2}, d = u_{2}^{2}, b = c = u_{1} u_{2}$

Compute, $a^{2} + b^{2} + c^{2} + d^{2}$

$a^{2} + b^{2} + c^{2} + d^{2} = {(u_{1}^{2})}^{2} + {(u_{1} u_{2})}^{2} + {(u_{1} u_{2})}^{2} + {(u_{2}^{2})}^{2} \Rightarrow a^{2} + b^{2} + c^{2} + d^{2} = (u_{1}^{2} + u_{2}^{2}) (u_{1}^{2} + u_{2}^{2})$

Hence, the statement is false.

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Linear Algebra With Applications

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Short Answer

TRUE OR FALSE?
If a matrix $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ represents the orthogonal projection onto a line L , then the equation $a^{2} + b^{2} + c^{2} + d^{2} = 1$ must hold.

Step by Step Solution

Step 1: Consider the parameters

Step 2: Find the values of the elements

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Linear Algebra With Applications

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TRUE OR FALSE?If a matrix A=(abcd) represents the orthogonal projection onto a line L , then the equation a2+b2+c2+d2=1 must hold.

Step by Step Solution

Step 1: Consider the parameters

Step 2: Find the values of the elements

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TRUE OR FALSE?
If a matrix $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ represents the orthogonal projection onto a line L , then the equation $a^{2} + b^{2} + c^{2} + d^{2} = 1$ must hold.