Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the dimension of the space $Q_{2}$ of all quadratic forms in two variables.

the solution is

$∵ \dim (Q_{2}) = 3$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identify the space's dimensions Q2

The space of all quadratic forms $q (x)$ in two variables is known as $Q_{2}$ . That's correct,

$Q_{2} = {q (x) = x^{T} Ax : A$

is a two-dimensional symmetric matrix. Each $q \in Q_{2}$ can be written as

$q (x) = x^{T} A x = [\begin{matrix} x_{1} & x_{2} \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = a x_{1}^{2} + 2 b x_{1} x_{2} + c c_{2}^{2}$

Where $a : b, c \in R$ and thus, dimension of

$Q_{2} = \{q (x_{1}, x_{2}) = a x_{1}^{2} + 2 b x_{1} x_{2} + c x_{2}^{2} : a, b, c \in R\}$ is This is the dimension of the order 2 symmetric matrix A .

$∴ \dim (Q_{2}) = 3$

Step 2: conclusion

$∴ \dim (Q_{2}) = 3$

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Linear Algebra With Applications

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Short Answer

Find the dimension of the space $Q_{2}$ of all quadratic forms in two variables.

Step by Step Solution

Step 1: Identify the space's dimensions Q2

Step 2: conclusion

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Linear Algebra With Applications

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Short Answer

Find the dimension of the space Q2 of all quadratic forms in two variables.

Step by Step Solution

Step 1: Identify the space's dimensions Q2

Step 2: conclusion

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Find the dimension of the space $Q_{2}$ of all quadratic forms in two variables.