§ 3

In the space of polynomials of degree $\leqslant 3$ we define the scalar product in the same way as in problem 1 of § 1. A linear function $\mathrm{f}$ assigns to a polynomial $p(t)$ its constant term $p(0)$. Find the vector (polynomial) associated with this linear function.

A linear transformation $\mathrm{A}$ is associated with a bilinear function $\mathrm{b}$. With which bilinear function is its adjoint transformation $\mathrm{A}^{*}$ associated?

In a basis $\mathbf{e}$ a bilinear function has matrix $B$. Find the matrix of its associated transformation if $\Gamma$ is the Gram matrix of the basis $\mathbf{e}$:

Prove that the value of a quadratic form $\mathrm{k}(x)$ on a vector $x$ of length 1 lies between the smallest and the largest eigenvalues of its associated transformation, and that these bounds are attained on the corresponding eigenvectors.

A quadratic form is given in an orthonormal basis by the polynomial

Find the change-of-basis matrix to an orthonormal basis in which it has diagonal form, and its form in this basis.

Let $\mathrm{k}$ and $\mathrm{h}$ be quadratic forms and let $\mathrm{h}$ be positive definite. Does there exist a basis in which $\mathrm{k}$ has canonical form and $\mathrm{h}$ — diagonal form?

Give an example of two quadratic forms which

Find the change-of-basis matrix to a basis in which the quadratic forms $\mathrm{k}(x)=\left(\xi^{1}\right)^{2}-2 \xi^{1} \xi^{2}+\left(\xi^{2}\right)^{2}$ and $\mathrm{h}(x)=17\left(\xi^{1}\right)^{2}+8 \xi^{1} \xi^{2}+\left(\xi^{2}\right)^{2}$ both have diagonal form, and also their form in this basis.

Prove that, in order that for two non-proportional quadratic forms in a two-dimensional space there exist a basis in which they both have diagonal form, it is necessary and sufficient that among their linear combinations there be a positive definite form. How essential is the assumption about the dimension of the space here?