Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider a quadratic form
$q (\overset{⇀}{x}) = \overset{⇀}{x} . A \overset{⇀}{x}$
where A is a symmetric nxn matrix. Find $q ({\overset{⇀}{e}}_{1})$ . Give your answer in terms of the entries of the matrix A.

$q ({\overset{⇀}{e}}_{1}) = a_{11}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information:

$q (\overset{⇀}{x}) = \overset{⇀}{x} . A \overset{⇀}{x}$

Step 2: Determining :

Take a look at the quadratic form:

$q (\overset{⇀}{x}) = \overset{⇀}{x} . A \overset{⇀}{x}$

Where A is a nxn symmetric matrix Take into account this:

$A = (a_{ij})$ $(where 1 \leq i, j \leq n)$

Now, use the following formula to access $q ({\overset{⇀}{e}}_{1})$ :

localid="1659613659189" $q ({\overset{⇀}{e}}_{1}) = {\overset{⇀}{e}}_{1} . A {\overset{⇀}{e}}_{1} = {\overset{⇀}{e}}_{1} (\sum_{j - 1}^{n} a_{j 1} {\overset{⇀}{e}}_{j}) = \sum_{j - 1}^{n} a_{j 1} ({\overset{⇀}{e}}_{i} . {\overset{⇀}{e}}_{j}) \Rightarrow q ({\overset{⇀}{e}}_{1}) = a_{11} (Since {\overset{⇀}{e}}_{1} . {\overset{⇀}{e}}_{j} = δ_{ij})$

Step 3: Determining the Result:

$q ({\overset{⇀}{e}}_{1}) = a_{11}$

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Linear Algebra With Applications

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Short Answer

Consider a quadratic form
$q (\overset{⇀}{x}) = \overset{⇀}{x} . A \overset{⇀}{x}$
where A is a symmetric nxn matrix. Find $q ({\overset{⇀}{e}}_{1})$ . Give your answer in terms of the entries of the matrix A.

Step by Step Solution

Step 1: Given Information:

Step 2: Determining :

Step 3: Determining the Result:

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Step by Step Solution

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Consider a quadratic form
$q (\overset{⇀}{x}) = \overset{⇀}{x} . A \overset{⇀}{x}$
where A is a symmetric nxn matrix. Find $q ({\overset{⇀}{e}}_{1})$ . Give your answer in terms of the entries of the matrix A.