§ 6

The value of a bilinear function $\mathrm{b}$ in some basis is written as a polynomial in the coordinates $\xi^{i}$ and $\eta^{j}$ of the vectors $x$ and $y$ :

Write the matrix of this bilinear function if the space is

How will the matrix of the bilinear function from problem $1(a)$ change if we pass to the basis $e_{1}^{\prime }=e_{1}+e_{2}, e_{2}^{\prime }=e_{2}+e_{3}, e_{3}^{\prime }=e_{3}$ ?

Write the matrix of the quadratic form $\left(\xi^{1}\right)^{2}+\xi^{1} \xi^{2}+\left(\xi^{2}\right)^{2}$.

Reduce to canonical form the quadratic form with matrix

The null space of a symmetric bilinear function $\mathrm{b}$ is the set of vectors $x$ such that $\mathrm{b}(x, y)=0$ for all $y$. Verify that this is a linear subspace. How is its dimension $s$ related to the rank of $\mathrm{b}$? What will the matrix of the function $\mathrm{b}$ be in a basis whose last $s$ vectors lie in the null space?

$B n$-dimensional space $m$ quadratic forms are given. Under what condition does there exist a basis in which they can all be represented as polynomials in the first $k<n$ coordinates of a vector?

Let $A$ be a square matrix of order $n$ and rank $r$. For the quadratic form with matrix $A^{T} A$ determine

A quadratic form with matrix $B$ is positive definite if and only if there exists a nonsingular upper triangular matrix $R$ such that $B=R^{T} R$. Prove this.

A quadratic form $\mathrm{k}$ is given. Under what condition does there exist a nonzero vector $x$ for which $\mathrm{k}(x)=0$ ?

What necessary and sufficient condition must the principal minors of a negative definite quadratic form satisfy?