Biorthogonal basis (31.33-31.42)

How does the biorthogonal basis transform if the given basis is transformed by the transition matrix $S$?

Construct a basis of the space $\mathcal{P}^{(2)}$ biorthogonal to the basis of functions $\varphi_{1}, \varphi_{2}, \varphi_{3}$ defined in problem 31. 23.

Find a basis of the space $\mathcal{P}^{(n)}$ biorthogonal to the basis of functions $\varphi_{1}, \varphi_{2}, \ldots , \varphi_{n + 1}$ constructed in problems $31. 21, 31. 22$. Compute the coordinates of an arbitrary polynomial in the basis found.

Construct a basis of the space $\mathcal{P}^{(2)}$ biorthogonal to the basis of functions $\delta_{0}, \delta_{1}, \delta_{2}$ defined in problem 31. 32.

Let $e_{1}, \ldots , e_{n}$ be a basis in the space $\mathcal{L}_{n}$, and let $\mathrm{f}_{1}, \ldots , \mathrm{f}_{n}$ be the basis in $\mathcal{L}_{n}^{ * }$ biorthogonal to it. Prove that for all $x\in \mathcal{L}_{n}$ the equality holds

and for all y $\in \mathcal{L}_{n}^{ * }$ the equality

Apply formula (1) to the bases considered in problem 31. 34.

Using the result of problem 31. 34, prove that polynomials $p_{0}, \ldots , p_{k}$ of degree at most $k$ are linearly independent if and only if for some $t_{0}$ we have $\operatorname {det}\left\| p_{2}^{(\jmath)}\left(t_{0}\right)\right\| \neq 0$.

Find the basis biorthogonal to the standard basis of the space $\mathcal{R}_{n \times n}$. Compute the matrices $C$ corresponding to the functions of this basis (in the sense of problem 31. 17).

The Pauli matrices

form a basis in the space of complex square matrices of order 2. Find the basis biorthogonal to the basis $\sigma_{0}, \sigma_{1}$, $\sigma_{2}, \sigma_{3}$, and compute the matrices $C$ corresponding to the functions of this basis in the sense of problem 31. 17.