Orthogonal Systems of Vectors

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.