Metric vector spaces

Which of the vector spaces with bilinear forms from problem 37.1 are metric?

Prove that the real part $f(x, y)$ and the imaginary part $g(x, y)$ of a Hermitian function on a complex vector space $V$ are invariant under multiplication by $i$, i.e., for any vectors $x, y \in V$

Prove that a metric vector space is the direct sum of a subspace $L$ and its orthogonal complement $L^{\perp }$ if and only if the inner product on $L$ is nondegenerate, and that in this case the inner product on $L^{\perp }$ is also nondegenerate.

In the space $\mathbf{M}_{n}(\mathbb {C})$ with Hermitian inner product

find the orthogonal complement of the subspace of:

Show that Hermitian and Euclidean spaces are normed.

Which of the norms on the spaces $\mathbb {R}^{n}, \mathbb {C}^{n}$ from problem 42.1 are induced by a Euclidean or Hermitian metric?

Complete the following system of vectors of a Euclidean or Hermitian space to an orthogonal basis:

Find the orthogonal projection of a vector $x$ of a Euclidean (Hermitian) space onto the linear span of an orthonormal system of vectors $\left(e_{1}, \ldots , e_{k}\right)$.

Prove that in any two subspaces of a Euclidean (Hermitian) space one can choose orthonormal bases $\left(e_{1}, \ldots , e_{k}\right)$ and $\left(f_{1}, \ldots , f_{l}\right)$ such that $\left(e_{i}, f_{j}\right)=0$ for $i \neq j$ and $\left(e_{i}, f_{i}\right) \geqslant 0$.

Let $\left(e_{1}, \ldots , e_{k}\right)$ and $\left(f_{1}, \ldots , f_{l}\right)$ be orthonormal bases of subspaces $L$ and $M$ of a Euclidean (Hermitian) space, and let $A=\left(\left(e_{i}, f_{j}\right)\right)$ be a matrix of order $k \times l$. Prove that all characteristic numbers of the matrix ${ }^{t} A \cdot A$ belong to the interval $[0,1]$ and do not depend on the choice of bases in the subspaces $L$ and $M$.

Prove that every real symmetric matrix of rank $\leqslant n$ with nonnegative (respectively positive) principal minors is the Gram matrix of some system (respectively linearly independent system) of vectors of an $n$-dimensional Euclidean space.

Prove the analogous statement for a Hermitian matrix and a Hermitian space.

Prove that the sum of the squares of the lengths of the projections of the vectors of any orthonormal basis of a Euclidean (Hermitian) space onto a $k$-dimensional subspace equals $k$.

Let $G$ be the matrix of the inner product in the basis $\left(e_{1}, \ldots , e_{n}\right)$ of a Euclidean space $V$. Find the transition matrix to the dual basis $\left(f_{1}, \ldots , f_{n}\right)$ and the matrix of the inner product in this basis.

Let $S$ be the transition matrix from the basis $\mathbf{e}$ to the basis $\mathbf{e}^{\prime }$. Find the transition matrix from the basis $\mathbf{e}^{\prime }$, dual to $\mathbf{e}$, to the basis $\mathbf{f}^{\prime }$, dual to $\mathbf{f}$:

Using the orthogonalization process, construct an orthogonal basis of the linear span of the system of vectors of a Euclidean (Hermitian) space:

Find a basis of the orthogonal complement of the linear span of the system of vectors of a Euclidean (Hermitian) space:

Prove that the systems of linear equations defining a linear subspace of $\mathbb {R}^{n}$ and its orthogonal complement are related as follows: the coefficients of a linearly independent system defining one of these subspaces are the coordinates of the basis vectors of the other subspace.

Find the equations defining the orthogonal complement of the subspace given by the system of equations:

Find the projection of the vector $x$ onto the subspace $L$ and the orthogonal component of the vector $x$:

Prove that if, in the process of orthogonalization, the system of vectors $a_{1}, \ldots , a_{n}$ passes into the system $b_{1}, \ldots , b_{n}$, then the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ relative to the linear span of the system $a_{1}, \ldots , a_{k-1}(k>1)$.

Find the distance from the vector $x$ to the subspace given by the system of equations:

Let $\left(e_{1}, \ldots , e_{k}\right)$ be an orthonormal system of vectors of an $n$-dimensional Euclidean (Hermitian) space. Prove that for any vector $x$ Bessel's inequality holds

and equality is attained for any $x$ if and only if $k=n$, i.e., the given system of vectors is an orthonormal basis of the space $V$ (Parseval's equality).

Using the Cauchy inequality, prove that

for any complex numbers $a_{1}, \ldots , a_{k}, b_{1}, \ldots , b_{k}$.

Prove that the square of the distance from a vector $x$ of a Euclidean (Hermitian) space to the subspace with basis $(e_{1}, \ldots , e_{k})$ equals the ratio of the Gram determinants of the systems of vectors $\left(e_{1}, \ldots , e_{k}, x\right)$ and $\left(e_{1}, \ldots , e_{k}\right)$.

Prove that the Gram determinant of any system of vectors:

Prove that the determinant of the matrix of a positive definite quadratic form does not exceed the product of the entries on its main diagonal.

Prove that for any real square matrix $A=\left(a_{i j}\right)$ of order $n$ Hadamard's inequality holds

with equality holding if and only if either

or the matrix $A$ has a zero row. Formulate and prove the analogous statement for a complex matrix $A$.

Find the lengths of the sides and the interior angles of triangle $A B C$ in the space $\mathbb {R}^{5}$:

Using the inner product of vectors, prove that:

Using the method of least squares, solve the overdetermined system of linear equations:

Prove that the square of the diagonal of an $n$-dimensional rectangular parallelepiped equals the sum of the squares of its edges emanating from one vertex.

Find the number of diagonals of an $n$-dimensional cube orthogonal to a given diagonal.

Find the length of the diagonal of an $n$-dimensional cube with edge $a$ and the angles between the diagonals of the cube and its edges.

Find the radius of the ball circumscribed about an $n$-dimensional cube with edge $a$, and solve the inequality $R<a$.

Prove that the length of the orthogonal projection of an edge of an $n$-dimensional cube onto any of its diagonals equals $1 / n$ of the length of the diagonal.

Compute the volume of the $n$-dimensional parallelepiped with sides:

Prove that for the volume of a parallelepiped the inequality holds

with equality holding if and only if $\left(a_{i}, b_{j}\right)=0$ for all $i$ and $j$.

Find the angle between the vector $x$ and the subspace $L$:

Prove that if every two distinct vectors among $k$ vectors of a Euclidean space $V$ form an angle of $\pi / 3$ with each other, then $k \leqslant \operatorname {dim} V$.

Prove that if every two distinct vectors among $k$ vectors of a Euclidean space form an obtuse angle, then $k \leqslant 1+\operatorname {dim} V$.

Find the angle between the diagonal of an $n$-dimensional cube and its $k$-dimensional face.

Find the angle between the two-dimensional faces $a_{0} a_{1} a_{2}$ and $a_{0} a_{3} a_{4}$ of a regular four-dimensional simplex $a_{0} a_{1} a_{2} a_{3} a_{4}$.

Find the angle between the subspaces

The polynomials

are called Legendre polynomials.

In the space $\mathbb {R}\left[x\right]_{n}$ with inner product

find:

In the space $L$ of continuous functions on the interval $[-\pi , \pi ]$ with inner product

find the projection of the function $t^{m}$ onto the subspace

Let $V$ be a pseudo-Euclidean space of signature $(p, q)$ and $W$ a subspace of $V$. Prove that:

Let a nondegenerate inner product of signature $(p, q)$ be given on a vector space, and suppose its restriction to a subspace $W$ is a nondegenerate inner product of signature $(p^{\prime }, q^{\prime })$. Prove that the restriction of the inner product to $W^{\perp }$ is nondegenerate and has signature $\left(p-p^{\prime }, q-q^{\prime }\right)$.

Prove that in a pseudo-Euclidean space of signature $(p, q)$, where $p$ and $q$ are both nonzero, there exists a basis consisting of isotropic vectors.

Prove the following properties of the operation of passing to the adjoint operator in a metric space:

Find the matrix of the operator $\mathcal{A}^{*}$ in the basis $\mathbf{e}$ of a metric vector space $V$, if the operator $\mathcal{A}$ has matrix $A$ in this basis and the inner product has matrix $G$.

Let $\left(e_{1}, e_{2}\right)$ be an orthonormal basis of a metric vector space and let the operator $\mathcal{A}$ have matrix $\left(\begin{array}{rr}1 & 2 \\ 1 & -1\end{array}\right)$ in the basis $\left(e_{1}, e_{1}+e_{2}\right)$. Find the matrix of the operator $\mathcal{A}^{*}$ in this basis.

Find the operator adjoint to the projection of the coordinate plane onto the abscissa axis parallel to the bisector of the first and third quadrants.

Let $\mathcal{A}$ be the projection of a metric vector space $V$ onto a subspace $V_{1}$ parallel to a subspace $V_{2}$. Prove that:

Prove that if a subspace of a metric vector space is invariant with respect to a linear operator $\mathcal{A}$, then its orthogonal complement is invariant with respect to the operator $\mathcal{A}^{*}$.

Prove that the kernel and image of the adjoint operator $\mathcal{A}^{*}$ are the orthogonal complements of the image and kernel of the operator $\mathcal{A}$, respectively.

Prove that if $x$ is an eigenvector of the operators $\mathcal{A}$ and $\mathcal{A}^{*}$ in a metric vector space with eigenvalues $\lambda$ and $\mu$, then $\mu =\bar{\lambda }$.

Let $V$ be the space of real infinitely differentiable periodic functions of period $h>0$ with inner product $\int_{0}^{h} f(x) g(x) d x$.

Let $V$ be the space of real infinitely differentiable functions on the interval $[a, b]$ with inner product $\int_{a}^{b} f(x) g(x) d x$. Prove that:

Prove that if the linear operators $\mathcal{A}$ and $\mathcal{B}$ in the space $\mathbb {R}\left[x\right]$ with inner product $\int_{a}^{b} f(x) g(x) d x$ are defined by the rules

where $P(x, y) \in \mathbb {R}\left[x, y\right]$, then $\mathcal{B}=\mathcal{A}^{*}$.

Prove that if $\mathcal{A}$ is a self-adjoint operator, then the function $f(x, y)=(\mathcal{A} x, y)$ is Hermitian.

Prove that if $\mathcal{A}$ and $\mathcal{B}$ are self-adjoint operators in a metric vector space $V$ and $(\mathcal{A} x, x)=(\mathcal{B} x, x)$ for all $x \in V$, then $\mathcal{A}=\mathcal{B}$.

Prove that an operator $\mathcal{A}$ in a Euclidean or Hermitian space $V$ is normal if and only if $\left|\mathcal{A} x\right|=\left|\mathcal{A}^{*} x\right|$ for all $x \in V$.

Prove that if $x$ is an eigenvector of a normal operator $\mathcal{A}$ in a Euclidean or Hermitian space with eigenvalue $\lambda$, then $x$ is an eigenvector of the operator $\mathcal{A}^{*}$ with eigenvalue $\bar{\lambda }$.

Prove that eigenvectors of a normal operator in a metric vector space with distinct eigenvalues are orthogonal.

Prove that:

Prove that any set of commuting normal operators in a Hermitian space has a common orthonormal basis.

Prove that if a normal operator $\mathcal{A}$ in a Hermitian space commutes with an operator $\mathcal{B}$, then $\mathcal{A}$ commutes with $\mathcal{B}^{*}$.

Let $\mathcal{A}, B$ be normal operators in a Hermitian space, with the characteristic polynomials of these operators equal. Prove that the matrices of the operators $\mathcal{A}$ and $B$ are similar in any basis.

Let $\mathcal{A}$ be a normal nilpotent operator in a Hermitian space. Then $\mathcal{A}=0$.

An operator $\mathcal{A}$ in a Hermitian space is normal if and only if $\mathcal{A}^{*}=p(\mathcal{A})$ for some polynomial $p(t)$.

For every polynomial

set

Prove that if $\mathcal{A}$ is an operator in a metric space, then:

Let $\mathcal{A}$ be a normal operator in a metric vector space $V$ and $f(x) \in K[x]$.

Prove that:

Let $\mathcal{A}$ be a normal operator in a Euclidean space $V$, with $\mathcal{A}^{2}=-E$. Prove that $\mathcal{A}^{*}=-\mathcal{A}$.

Let $p(t)=t^{2}+a t+b$ be a real irreducible polynomial. Suppose that $\mathcal{A}$ is a normal operator in a Euclidean space, with $p(\mathcal{A})=0$. Prove that $A^{*}=-\mathcal{A}-a E$.