Spaces with a Scalar Product

Prove that the requirement of bilinearity (sesquilinearity) in the axioms of a Euclidean (Hermitian) space can be replaced by linearity in the second argument.

Show that the function $f$ of the vectors $x$ and $y$ of the real plane, given by the formula

defines a Euclidean inner product on the plane. Find the inner product of the vectors $(1,-1)$ and $(4,1)$.

Consider the function $f$ of the vectors $x$ and $y$ of a two-dimensional space, defined as follows:

Prove that this formula defines a Euclidean inner product if and only if $a > 0, a d-b^{2} > 0$.

Find the lengths of the following vectors of the Hermitian space $\mathbb {C}^{3}$:

Check whether the following bilinear functions on the space of polynomials of degree at most $n$ are Euclidean inner products:

The polynomials $x^{2}$ and $x^{4}$ are given. Find their lengths and the angle $\varphi$ between them in the Euclidean space of polynomials of degree at most 4 with the inner product given by the integral:

Check whether:

Using the Cauchy--Bunyakovsky--Schwarz inequality, prove the following inequalities:

Using the Cauchy--Bunyakovsky--Schwarz inequality, prove that the triangle inequalities

hold in a Euclidean (Hermitian) space, where the right-hand inequality becomes an equality if and only if the vectors $x$ and $y$ are collinear and equally directed.

(Pythagorean theorem). Prove that if the vectors $x$ and $y$ in a Euclidean (Hermitian) space are orthogonal, then

Is the converse statement true?

Prove that in the parallelogram spanned by the vectors $x$ and $y$ of a Euclidean space, the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides. Is the analogous statement true for a Hermitian space?

Prove that every subspace $L$ of a Euclidean (Hermitian) space $V$ is a Euclidean (Hermitian) space with respect to the restriction to $L$ of the inner product from $V$, i.e., the inner product defined as follows: the inner product $\left(l_{1}, l_{2}\right)_{L}$ of two vectors $l_{1}, l_{2} \in L$ equals their inner product $\left(l_{1}, l_{2}\right)_{V}$ as vectors of $V$. Such an inner product is called the inner product induced by the embedding $L \subset V$ of the subspace $L$ in the space $V$.

On the external direct sum $V \oplus W$ of two Euclidean (Hermitian) spaces $V$ and $W$, introduce a Euclidean (Hermitian) inner product so that the inner products on $V$ and $W$ induced by the standard embeddings

coincide with their original inner products $(\cdot , \cdot )_{V}$ and $(\cdot , \cdot )_{W}$ respectively, and so that the subspaces $V$ and $W$ are orthogonal to each other.

Prove the Cauchy--Bunyakovsky--Schwarz inequality for a Hermitian space:

Let $V$ be a real Euclidean space with inner product $(\cdot , \cdot )_{\mathbb {R}}$. Show that its complexification $V_{\mathbb {C}}$ can be turned into a Hermitian space by introducing on it the Hermitian inner product $(\cdot , \cdot )_{\mathbb {C}}$ by the formula

The space $V_{\mathbb {C}}$ with the Hermitian product $(\cdot , \cdot )_{\mathbb {C}}$ defined in this way will be called the complexification $V_{\mathbb {C}}$ of the Euclidean space $V$. Which Hermitian space is obtained by complexifying the standard $\mathbb {R}^{n}$?

Prove that if $e_{1}, e_{2}, \ldots , e_{n}$ is an orthonormal basis in a real Euclidean space $V$, then the vectors $e_{1}+i \cdot 0$, $e_{2}+i \cdot 0, \ldots , e_{n}+i \cdot 0$ form an orthonormal basis in the complexification $V_{\mathbb {C}}$ with respect to the Hermitian inner product introduced in the previous problem.

Let $W$ be a Hermitian space with Hermitian inner product $(\cdot ,\cdot )$. Consider its realification $W_{\mathbb {R}}$. Prove that $W_{\mathbb {R}}$ becomes a Euclidean space if one introduces on $W_{\mathbb {R}}$ the Euclidean inner product $(\cdot , \cdot )_{\mathbb {R}}$ by the formula

In the case $W = \mathbb {C}^{n}$, write the formula for the value $\left(z_{1}, z_{2}\right)_{\mathbb {R}}$ in the standard coordinates $\left(x^{1}, \ldots , x^{n}, y^{1}, \ldots , y^{n}\right)$ of the realified space $W_{\mathbb {R}}$, given by the equalities $x^{k}+i y^{k} = z^{k}$. Show that if the standard Hermitian space $\mathbb {C}^{n}$ is realified in this way, the result is the standard Euclidean space $\mathbb {R}^{2 n}$.

Show that, under the conditions of the previous problem, the vectors $z$ and $i z$ are always orthogonal with respect to $(\cdot , \cdot )_{\mathbb {R}}$.

Problem 1284: $W$ is a Hermitian space with Hermitian inner product $(\cdot , \cdot )$, $W_{\mathbb {R}}$ is its realification with the Euclidean inner product $\left(z_{1}, z_{2}\right)_{\mathbb {R}} = \operatorname {Re}\left(z_{1}, z_{2}\right)$.

Show that the imaginary part $\operatorname {Im}(\cdot , \cdot )$ of a Hermitian inner product is a real skew-symmetric bilinear function on the Hermitian space $W$, and that the form $\operatorname {Im}(\cdot , \cdot )$ is invariant under multiplication by $i$, i.e.,

Show that the Hermitian inner product $(\cdot , \cdot )$ is uniquely recovered both from its real part $\operatorname {Re}(\cdot , \cdot )$ and from its imaginary part $\operatorname {Im}(\cdot , \cdot )$. To this end, express:

Prove that every orthogonal system of nonzero vectors in a Euclidean (Hermitian) space is linearly independent.

Check the orthogonality of the given vectors and extend them to an orthogonal basis of the entire space $\mathbb {R}^{4}:(1,-1,-1,1)$, $(3,1,1,-1)$.

Check the orthogonality of the pair of vectors $(1,-i, 1+i$), $(1+i,-1+i, 0)$ of the Hermitian space $\mathbb {C}^{3}$ and extend it to an orthogonal basis of the entire space $\mathbb {C}^{3}$.

With respect to the Euclidean inner product $(A, B) = \operatorname {tr}\left(A^{\top } B\right) = \sum_{i, j = 1}^{n} a_{i j} b_{i j}$, check the orthogonality of the following system of matrices:

and extend it to an orthogonal basis of the entire space of real square matrices of order two.

Extend the given system of vectors to an orthonormal basis of the entire three-dimensional space:

Extend the system of vectors to an orthonormal basis of the entire space $\mathbb {R}^{4}$:

Extend the given system of vectors to an orthonormal basis of the entire Hermitian space $\mathbb {C}^{4}$:

Using the Gram-Schmidt orthogonalization method, construct an orthogonal basis of the subspace spanned by the vectors:

Using the Gram-Schmidt orthogonalization method, construct an orthogonal basis of the subspace of the space of polynomials spanned by the polynomials $x^{3}, x^{4}, x^{5}, x^{6}$ with the inner product given by the integral:

Using the Gram-Schmidt orthogonalization method, construct an orthogonal basis of the Hermitian space $\mathbb {C}^{3}$ spanned by the vectors $(1+i, 0,1-i),(1+i,-i, 1+i),(-2 i, 1,2)$.

Prove that the orthogonalization process does not increase the lengths of the vectors being orthogonalized, i.e.,

where the vectors $b_{k}$ are obtained from the system $a_{k}$ by the orthogonalization process. Under what conditions does the equality $\left|b_{l}\right| = \left|a_{l}\right|$ hold for some $l, 1 < l \leqslant n$?

Prove that a zero vector $b_{l}$ arises in the orthogonalization process applied to a system of vectors $a_{k}$ of a Euclidean space if and only if the original system $a_{k}$ was linearly dependent. More precisely, $b_{l} = 0$ if and only if the vector $a_{l}$ is a linear combination of the vectors $a_{i}, i = 1,2, \ldots , l-1$.

A sequence of polynomials is given by the recurrence formula

A sequence of polynomials is given by the recurrence formula

Find explicit expressions for $P_{2}(x), P_{3}(x), P_{4}(x), P_{5}(x)$.

The polynomials defined in this way are called the Legendre polynomials.

Prove that the polynomials defined by the formulas

Prove that the orthogonal basis $Q_{k}(t), k = 1,2, \ldots , n$, in the space of polynomials of degree at most $n$ with the inner product given by the integral

can be obtained from the Legendre polynomials $P_{k}(x)$ by some linear change of variable $x = x(t)$ (i.e., $Q_{k}(t) = P_{k}(x(t))$).

Prove that the transition matrix from the basis $a_{1}, a_{2}, \ldots , a_{k}$ to the basis $b_{1}, b_{2}, \ldots , b_{k}$, obtained from the basis $a_{1}, a_{2}, \ldots , a_{k}$ by the orthogonalization process, is always upper triangular, with all diagonal entries of this matrix equal to one.

Let a linearly independent system of vectors $a_{1}$, $a_{2}, \ldots , a_{k}$ and an orthogonal system $c_{1}, c_{2}, \ldots , c_{k}$ be given such that the transition matrix from the first system to the second is upper triangular. Prove that the vectors $b_{i}$ of the orthogonal system obtained from $a_{1}, a_{2}, \ldots , a_{k}$ by the orthogonalization process differ from the vectors of the system $c_{1}, c_{2}, \ldots , c_{k}$ only by constant factors: $b_{i} = \alpha_{i} c_{i}$, $i = 1,2, \ldots , k$.

Prove that applying the orthogonalization process to the polynomials $1, x, x^{2}, \ldots , x^{n}$ in the Euclidean space $\mathbb {R}_{n}[x]$ of polynomials of degree at most $n$ with the inner product $(f, g) = \int_{-1}^{1} f(x) g(x) d x$ yields polynomials that differ from the Legendre polynomials $P_{k}(x)$ only by constant factors. Find these factors.

An orthogonal (unitary) matrix $U$ is the transition matrix from one orthonormal basis to another in a Euclidean (Hermitian) space. Prove that:

Prove that an arbitrary nondegenerate matrix $A$ can be represented as the product $A = U R$ of an orthogonal (unitary) matrix $U$ and an upper triangular matrix $R$. To what extent is such a representation unique?

Represent the matrix

as the product of an orthogonal matrix $U$ and an upper triangular matrix $R$ with positive numbers on the diagonal.

Represent the matrix

as the product of a unitary matrix $U$ and an upper triangular matrix $R$.

Prove that the coordinates $x^{i}, i = 1, \ldots , n$, of an arbitrary vector $x$ with respect to an orthogonal basis $e_{1}, \ldots , e_{n}$ can be found by the formulas

Such coordinates are called the Fourier coefficients of the vector $x$ with respect to the orthogonal basis $e_{1}, \ldots , e_{n}$.

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

check the pairwise orthogonality of the functions of the following system:

$1, \cos x, \sin x, \cos 2 x, \sin 2 x, \ldots , \cos n x, \sin n x$.

In the subspace spanned by the functions

of the Euclidean space of trigonometric polynomials with the inner product

find the Fourier coefficients of the function $\cos^{3} x$ with respect to the orthogonal basis $f_{0}, f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}$.

Let $e_{1}, e_{2}, \ldots , e_{k}$ be an orthonormal system of vectors of the Euclidean space $\mathbb {R}^{n}$. Prove that Bessel's inequality

holds for all $x \in \mathbb {R}^{n}$, where equality (Parseval's equality) is attained for all $x$ if and only if the system $e_{1}, e_{2}, \ldots , e_{k}$ is an orthonormal basis of $\mathbb {R}^{n}$, i.e., $k = n$.

Show that the number $(e_{i}, x)$ equals the length of the projection of the vector $x$ onto the one-dimensional space spanned by the vector $e_{i}$, and that if the system $e_{1}, e_{2}, \ldots , e_{n}$ forms a basis of $\mathbb {R}^{n}$, then $\left(e_{i}, x\right)$ is simply equal to the $i$-th coordinate $x^{i}$ of the vector $x$ with respect to this basis.

Find the Gram matrix of the system of vectors $1, x, x^{2}$ in the Euclidean space of polynomials of degree at most 2 with the inner product defined as follows:

Prove that the Gram matrix $G\left(a_{1}, a_{2}, \ldots , a_{n}\right) = G$ of a system of vectors $a_{1}, a_{2}, \ldots , a_{n}$ of a Hermitian space is Hermitian, i.e., $G = \bar{G}^{\top }$.

Prove that a system of vectors $a_{1}, a_{2}, \ldots , a_{k}$ is linearly dependent if and only if the Gram determinant $g\left(a_{1}, a_{2}, \ldots , a_{k}\right)$ of this system equals zero.

Two systems of vectors $x_{1}, x_{2}, \ldots , x_{n}$ and $y_{1}, y_{2}, \ldots , y_{n}$ are called dual (reciprocal) if

Prove that each of two dual systems of vectors is linearly independent.

Prove that for any basis of a Euclidean space there exists a unique dual basis. If the original basis is orthonormal, then its dual basis coincides with it.

Let two dual systems of vectors $x_{1}, x_{2}, \ldots , x_{n}$ and $y_{1}$, $y_{2}, \ldots , y_{n}$ be given. Prove that for $1 \leqslant k \leqslant n-1$ the system of vectors $y_{k+1}, \ldots , y_{n}$ is a basis of the orthogonal complement of the linear span of the system $x_{1}, x_{2}, \ldots , x_{k}$.

Find the dual basis for the system $1, x, x^{2}$ in the Euclidean space of polynomials of degree at most 2 with the inner product defined as follows:

A basis $e_{1}, e_{2}, \ldots , e_{n}$ of a Euclidean space and its Gram matrix $G = G\left(e_{1}, e_{2}, \ldots , e_{n}\right)$ are given. Find the transition matrix from the basis $e_{1}, e_{2}, \ldots , e_{n}$ to its dual basis.

Let $S$ be the transition matrix from the basis $e_{1}, e_{2}, \ldots , e_{n}$ to the basis $e_{1}^{\prime }, e_{2}^{\prime }, \ldots , e_{n}^{\prime }$. Find the transition matrix $T$ from the basis $f_{1}, f_{2}, \ldots , f_{n}$, dual to $e_{1}, e_{2}, \ldots , e_{n}$, to the basis $f_{1}^{\prime }, f_{2}^{\prime }, \ldots , f_{n}^{\prime }$, dual to $e_{1}^{\prime }, e_{2}^{\prime }, \ldots , e_{n}^{\prime }$:

Prove that the square of the length of the orthogonal component $z$ of a vector $x$ with respect to a space $L$ given by its basis $a_{1}, a_{2}, \ldots , a_{k}$, can be computed using the Gram determinant:

Prove that the Gram determinant $g\left(a_{1}, a_{2}, \ldots , a_{k}\right)$ of a system of vectors $a_{1}, a_{2}, \ldots , a_{k}$ does not change under the orthogonalization process, i.e.,

where the vectors $b_{1}, b_{2}, \ldots , b_{k}$ are obtained from the system $a_{1}, a_{2}, \ldots , a_{k}$ by the orthogonalization process.

Prove that all principal minors of the Gram matrix of a linearly independent system of vectors are always positive, and, in the case of a linearly dependent system, nonnegative.

Express, in terms of the Gram determinants $g_{0} = 1, g_{1}\left(a_{1}\right), \ldots \ldots , g_{n}\left(a_{1}, \ldots , a_{n}\right)$ of the subsystems of the system of vectors $a_{1}, \ldots , a_{n}$, the squares of the lengths $\left|b_{k}\right|^{2}, k = 1, \ldots , n$, of the vectors obtained from $a_{1}, \ldots , a_{n}$ by the orthogonalization process.

Prove that the formulas of the orthogonalization process (up to multiplication of the orthogonal vectors $b_{i}$ by certain constants $\alpha_{i}$, i.e., the formulas for $b_{i}^{\prime } = \alpha_{i} b_{i}$) can be rewritten in the form

Prove that the $n$-th Legendre polynomial (see problem 1302) equals, up to some numerical factor $c_{n}$, the following expression:

where $g\left(1, x, \ldots , x^{k-1}, x^{k+1}, \ldots , x^{n}\right)$ is the Gram determinant of the system of vector-polynomials obtained by removing the polynomial $x^{k}$ from the standard basis $1, x, x^{2}, \ldots , x^{n}$ of polynomials of degree at most $n$. Recall that the inner product is given by the formula

for arbitrary polynomials $f_{1}$ and $f_{2}$.

Prove that for the Gram determinant $g\left(a_{1}, a_{2}, \ldots , a_{k}\right)$ the following inequality (Hadamard's inequality) holds

where equality holds if and only if the vectors $a_{1}, a_{2}, \ldots , a_{k}$ are pairwise orthogonal or at least one of them is zero.

Prove that a real symmetric matrix $A = \left(a_{i j}\right)$ is the Gram matrix of some system of vectors $e_{1}, e_{2}, \ldots , e_{n}$ of a Euclidean space, i.e., $a_{i j} = \left(e_{i}, e_{j}\right), i, j = 1,2, \ldots , n$, if and only if all principal minors of the matrix $A$ are nonnegative.

Prove that any Hermitian matrix $A = \left(a_{i j}\right)$ with nonnegative principal minors is the Gram matrix of some system of vectors $e_{1}, e_{2}, \ldots , e_{n}$ of a Hermitian space $V: a_{i j} = \left(e_{i}, e_{j}\right), i, j = 1,2, \ldots , n$.

Can the matrix

be the Gram matrix of:

Can the matrix

be the Gram matrix of some linearly independent system of vectors?

Can the matrix

be the Gram matrix of some system of vectors (not necessarily linearly independent)?

Prove that the definition of the $n$-dimensional volume is well-posed (i.e., that it does not depend on the choice of order in the system of vectors $a_{1}, \ldots , a_{n}$), by relating it to the Gram determinant $g\left(a_{1}, \ldots , a_{n}\right)$ via the formula

Prove that if the vectors $a_{1}, \ldots , a_{n}$ are given by their coordinates in some orthonormal basis, then the volume $V\left(a_{1}, a_{2}, \ldots , a_{n}\right)$ of the parallelepiped built on the vectors $a_{1}, a_{2}, \ldots , a_{n}$, can be found by the formula

where $\det A$ is the determinant of the matrix whose columns (or rows) contain the coordinates $\left\{ a_{i}^{j}\right\}$ of the vectors $a_{1}, \ldots , a_{n}$.

Determine the geometric meaning of Hadamard's inequality

and of its becoming an equality.