Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q41E (page 402)

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

Short Answer

Expert verified

the solution is

$鈭� \dim (Q_{2}) = 3$

Step by step solution

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 $q 鈭� Q_{2}$ 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 $a : b, c 鈭� R$ and thus, dimension of

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

$鈭� \dim (Q_{2}) = 3$

conclusion

$鈭� \dim (Q_{2}) = 3$

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91影视

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

Short Answer

Step by step solution

Identify the space's dimensions Q2

conclusion

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91影视

Find the dimension of the spaceQ2 of all quadratic forms in two variables.

Short Answer

Step by step solution

Identify the space's dimensions Q2

conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Pure Maths

Statistics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

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