Math Problems Vault

Problem 25.1

Let $\mathbf{n}$ be a fixed nonzero vector in geometric space. To an arbitrary pair of vectors $\mathbf{x}, \mathbf{y}$ let us assign:

?

(1)

the mixed product $(\mathbf{n}, \mathbf{x}, \mathbf{y})$

(2)

the inner product $(\mathbf{x} + \mathbf{n}, \mathbf{y} + \mathbf{n})$

(3)

$(\mathbf{n}, \mathbf{x})(\mathbf{n}, \mathbf{y})$ ;

(4)

$\left|\mathbf{n}\right|(\mathbf{x}, \mathbf{y})$ ;

(5)

$\left|\mathbf{x}\right|\left|\mathbf{y}\right|$ .

Can such a function be taken as an inner product?

Problem 25.2

Can the inner product in a real $n$ -dimensional vector space be given by the following function of the coordinates of the vectors:

?

(1)

$x_{1} y_{1} + 2x_{2} y_{2}$ ,

(1) $n = 2$ ;

(2) $n = 3$

(2)

$x_{1} y_{1} + x_{2}, \quad n = 2$

(3)

$3x_{1} y_{1} + 2x_{1} y_{2} + x_{2} y_{1} + x_{2} y_{2}, \quad n = 2$

(4)

$2x_{1} y_{1} + x_{1} y_{2} + x_{2} y_{1} + 2x_{2} y_{2}, \quad n = 2$

(5)

$x_{1} y_{1} + 2x_{1} y_{2} + 2x_{2} y_{1} + x_{2} y_{2} + x_{3} y_{3}, \quad n = 3$ .

Problem 25.3

An ellipse with semi-axes 2 and 1 is drawn in the plane. Let a vector $\mathbf{x}$ be given. Consider the vector $\mathbf{x}_{0}$ , codirectional with $\mathbf{x}$ , whose start is the center of the ellipse and whose end lies on the ellipse. Set $\varphi (\mathbf{x}) = \left|\mathbf{x}\right| / \left|\mathbf{x}_{0}\right|$ , and $\varphi (\mathbf{o}) = 0$ . Now to an arbitrary pair of vectors one can assign the number

F(\mathbf{x}, \mathbf{y}) = \varphi ^{2}(\mathbf{x} + \mathbf{y}) - \varphi ^{2}(\mathbf{x} - \mathbf{y}).

Prove that this defines an inner product. Find its expression in terms of the coordinates of the vectors in the canonical coordinate system of the ellipse.

?

Problem 25.4

(1)(2)(3)

?

(1)

Prove that the function $F(X, Y) = \operatorname {tr} X^{T} Y$ is written in terms of the entries of the matrices by formula (1) of the introduction, and can be taken as an inner product in the space of real matrices of size $m \times n$ .

(2)

The Euclidean norm of a matrix is called its length with respect to this inner product. Prove that the Euclidean norm equals the square root of the sum of the squares of all entries of the matrix.

(3)

Find the lengths of the vectors of the standard basis and the angles between them with respect to this inner product.

Problem 25.5

(1)(2)(3)

Consider the space of square matrices of order $n$ , and to each pair of matrices assign a number:

?

(1)

$F(X, Y) = \operatorname {tr} XY$

(2)

$F(X, Y) = \operatorname {tr} X\operatorname {tr} Y$

(3)

$F(X, Y) = \operatorname {det} XY$ .

Can such a function be taken as an inner product?

Problem 25.6

Let $P$ be a fixed square matrix of order $m$ . Under what conditions on this matrix is the function $F(X, Y) = \operatorname {tr} X^{T} PY$ an inner product in the vector space of matrices of size $m \times n$ ?

?

Problem 25.7

In the vector space of functions continuous on the segment $\left[ - 1, 1\right]$ , to the functions $f$ and $g$ let us assign the number

(f, g) = \int _{ - 1}^{1} f(t) g(t) dt.

Prove that this defines an inner product.

?

Problem 25.8

(1)(2)(3)(4)

In the vector space of polynomials of degree at most $n$ , to two polynomials $p$ and $q$ a number $F(p, q)$ is assigned. Prove that this defines an inner product:

?

(1)

$F(p, q) = \int_{ - 1}^{1} p(t) q(t) dt$

(2)

$F$ is the sum of products of the derivatives of order $k$ , evaluated at the point $t_{0}$ ,

F(p, q) = \sum _{k = 0}^{n} p^{(k)}\left(t_{0}\right) q^{(k)}\left(t_{0}\right)

(3)

$F$ is the sum of products of the coefficients of equal powers,

F(p, q) = \sum _{k = 0}^{n} \alpha _{k} \beta _{k}

(4)

$F$ is the sum of products of the values of $p$ and $q$ at $m > n$ distinct points,

F(p, q) = \sum _{k = 1}^{m} p\left(t_{k}\right) q\left(t_{k}\right)

. (Verify that the requirement $m > n$ is necessary.)

Problem 25.9

Prove that in the Euclidean space of problem $25. 8, 1)$ the Legendre polynomials

P_{0}(t) = 1, \quad P_{k}(t) = \frac{1}{2^{k} k!} \frac{d^{k}}{dt^{k}}\left(t^{2} - 1\right)^{k} \quad (k = 1, \ldots , n)

form an orthogonal basis. Find the lengths (norms) of these polynomials.

?

Problem 25.10

Let $\mathbf{e}$ be a basis in a vector space $\mathcal{E}$ . Prove that there exists in $\mathcal{E}$ one and only one inner product with respect to which the basis $\mathbf{e}$ is orthonormal.

?

Problem 25.11

Let two inner products $(x, y)_{1}$ and $(x, y)_{2}$ be given in a real vector space. Prove that for any positive numbers $\lambda$ and $\mu$ the function $(x, y) = \lambda (x, y)_{1} + \mu (x, y)_{2}$ is also an inner product.

?

Problem 25.12

A vector space $\mathcal{L}$ is given as the direct sum of subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ , in which inner products $(x, y)_{1}$ and $(x, y)_{2}$ are given. Let $x = x_{1} + x_{2}$ and $y = y_{1} + y_{2}$ , where $x_{1}, y_{1} \in \mathcal{L}_{1}; x_{2}, y_{2} \in \mathcal{L}_{2}$ . Prove that the function $(x, y) = \left(x_{1}, y_{1}\right)_{1} + \left(x_{2}, y_{2}\right)_{2}$ is an inner product on $\mathcal{L}$ .

?

Problem 25.13

(1)(2)

Let $\left|x\right|$ be the length of a vector $x$ in a Euclidean space. Prove that:

?

(1)

$(x, y) = \frac{1}{4}\left(\left|x + y\right|^{2} - \left|x - y\right|^{2}\right)$

(2)

$(x, y) = \frac{1}{2}\left(\left|x + y\right|^{2} - \left|x\right|^{2} - \left|y\right|^{2}\right)$ .

Problem 25.14

Let two inner products $(x, y)_{1}$ and $(x, y)_{2}$ be given in a real vector space, and let every vector have the same length in each of them: $(x, x)_{1} = (x, x)_{2}$ . Prove that the inner products coincide.

?

Problem 25.15

(1)(2)

In the space of polynomials of degree $\leqslant 3$ with the standard inner product, a triangle is given with sides $t, t^{3}$ and $t - t^{3}$ . Find

?

(1)

the angles of the triangle

(2)

the lengths of its sides.

Problem 25.16

Prove that a triangle in a Euclidean space is right-angled if and only if the length of one of the sides equals the length of another side multiplied by the cosine of the angle between these sides.

?

Problem 25.17

Prove that a median of a triangle in a Euclidean space is shorter than one of the sides between which it lies.

?

Problem 25.18

Prove that the sum of the angles of an arbitrary triangle in a Euclidean space equals $\pi$ .

?

Problem 25.19

Let $\mathcal{L}$ be a finite-dimensional vector space. Prove that choosing a specific isomorphism of $\mathcal{L}$ onto its dual $\mathcal{L}^{ * }$ is equivalent to giving an inner product in $\mathcal{L}$ .

?