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Q41E

Expert-verified
Found in: Page 402

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

Find the dimension of the space ${{\mathbf{Q}}}_{{\mathbf{2}}}$ of all quadratic forms in two variables.

the solution is

$\because \mathrm{dim}\left({\mathrm{Q}}_{2}\right)=3$

See the step by step solution

Step 1:  Identify the space's dimensions Q2

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

${Q}_{2}=\left\{q\left(\mathrm{x}\right)={x}^{T}\mathrm{Ax}:A$

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

$q\left(x\right)={x}^{T}Ax=\left[\begin{array}{cc}{x}_{1}& {x}_{2}\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]\phantom{\rule{0ex}{0ex}}=a{x}_{1}^{2}+2b{x}_{1}{x}_{2}+c{c}_{2}^{2}$

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

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

$\therefore \mathrm{dim}\left({\mathrm{Q}}_{2}\right)=3$

Step 2: conclusion

$\therefore \mathrm{dim}\left({\mathrm{Q}}_{2}\right)=3$