Math Problems Vault

Problem 33.1

Verify that an $n$ -dimensional vector space $\mathcal{L}$ is an affine space with the space of vectors coinciding with $\mathcal{L}$ , if the points of this affine space are taken to be the vectors from $\mathcal{L}$ , and to every ordered pair of vectors $a, b$ there is assigned the vector $x = b - a$ .

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Problem 33.2

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

Prove that in an affine space $\mathcal{A}$ :

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(1)

$\overrightarrow {AA} =$ o for any point $A$ from $\mathcal{A}$

(2)

$P(A, o) = A$ for any point $A$ from $\mathcal{A}$

(3)

$\overrightarrow {AB} = - \overrightarrow {BA}$ for any points $A$ and $B$ from $\mathcal{A}$

(4)

the equality $\overrightarrow {AB} = \overline{A_{1} B_{1}}$ holds if and only if $\overrightarrow {AA_{1}} = \overrightarrow {BB_{1}}$ .

Problem 33.3

(1)(2)

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(1)

Prove that a system of points $A_{0}, A_{1}, \ldots , A_{k}$ of an affine space is independent if and only if there does not exist a plane of dimension less than $k$ containing this system of points.

(2)

Prove that a system of points $A_{0}, A_{1}, \ldots , A_{k}$ of an affine space is independent if and only if, for an arbitrary point $O$ , the equalities

\begin{gathered} \lambda _{0} \overrightarrow {OA_{0}} + \lambda _{1} \overrightarrow {OA_{1}} + \ldots + \lambda _{k} \overrightarrow {OA_{k}} = o, \\ \lambda _{0} + \lambda _{1} + \ldots + \lambda _{k} = 0 \end{gathered}

imply that $\lambda_{0} = \lambda_{1} = \ldots = \lambda_{k} = 0$ .

Problem 33.4

(1)(2)(3)

Is the system of points with coordinates independent:

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(1)

$(0, 1, 1), (1, 0, 1), (1, 1, 0)$

(2)

$(0, 1, 1), \quad (1, 0, 1), \quad (1, 1, 0), \quad (2, 2, 2)$

(3)

$(0, 1, 1), (1, 0, 1), \quad (1, 1, 0), \quad (2 / 3, 2 / 3, 2 / 3)$ ?

Problem 33.5

Show that the notion of independence of a system of points $A_{0}, A_{1}, \ldots , A_{k}$ is symmetric with respect to all the points of this system. Namely, if the system of vectors $\overrightarrow {A_{0} A_{1}}, \overrightarrow {A_{0} A_{2}}, \ldots , \overrightarrow {A_{0} A_{k}}$ is linearly independent, then any system $\overrightarrow {A_{j} A_{0}}, \ldots , \overrightarrow {A_{j} A_{j - 1}}, \overrightarrow {A_{j} A_{j + 1}}, \ldots , \overrightarrow {A_{j} A_{k}}, j = 1, 2, \ldots , k$ is also linearly independent.

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Problem 33.6

(1)(2)

Let $\mathrm{m}$ and $\mathrm{m}^{\prime }$ be planes with directing subspaces $\mathcal{M}$ and $\mathcal{M}^{\prime }$ . Prove that:

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(1)

if $\mathcal{M} \subset \mathcal{M}^{\prime }$ , then either $\mathrm{m}$ and $\mathrm{m}^{\prime }$ have no common points, or $\mathrm{m} \subset \mathrm{m}^{\prime }$

(2)

if $\mathcal{M} = \mathcal{M}^{\prime }$ , then $\mathrm{m}$ and $\mathrm{m}^{\prime }$ either have no common points or coincide.

Problem 33.7

Prove that if a line has two distinct common points with a plane, then it belongs to that plane.

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Problem 33.8

Prove that if a $k$ -dimensional plane $\mathrm{m}_{1}$ contains an independent system of points $A_{0}, A_{1}, \ldots , A_{k}$ common with a plane $\mathrm{m}_{2}$ , then $\mathrm{m}_{1} \subset \mathrm{m}_{2}$ .

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Problem 33.9

Prove that there exists exactly one $k$ -dimensional plane containing an independent system of points $A_{0}, A_{1}, \ldots , A_{k}$ .

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Problem 33.10

Let $A_{0}, A_{1}, \ldots , A_{k}$ be an independent system of points in a $k$ -dimensional plane $\mathrm{m}$ , and let $\mathrm{O}$ be a fixed point of the affine space. Prove that $\mathrm{m}$ consists of exactly those points $A$ for which

\overrightarrow {OA} = \lambda _{0} \overrightarrow {OA_{0}} + \lambda _{1} \overrightarrow {OA_{1}} + \ldots + \lambda _{k} \overrightarrow {OA_{k}}

where $\lambda_{0}, \lambda_{1}, \ldots , \lambda_{k}$ are numbers satisfying the equality $\lambda_{0} + \lambda_{1} + \ldots + \lambda_{k} = 1$ .

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Problem 33.11

Let $l_{1}, l_{2}, l_{3}, l_{4}$ be lines in an affine space, with $l_{1}$ parallel to $l_{2}$ , and $l_{3}$ parallel to $l_{4}$ . Let, further, $l_{3}$ intersect $l_{1}$ and $l_{2}$ at the points $A_{1}$ and $B_{1}$ respectively, and let $l_{4}$ intersect $l_{1}$ at the point $A_{2}$ . Prove that $l_{4}$ also intersects $l_{2}$ at a point $B_{2}$ such that $\overrightarrow {A_{1} A_{2}} = \overrightarrow {B_{1} B_{2}}, \overrightarrow {A_{1} B_{1}} = \overrightarrow {A_{2} B_{2}}$ .

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Problem 33.12

Prove that any two lines of an $n$ -dimensional affine space $(n \geqslant 3)$ are entirely contained in some three-dimensional plane.

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Problem 33.13

Under what necessary and sufficient condition are two lines $x = a_{0} + a_{1} t$ and $x = b_{0} + b_{1} t$ contained in one two-dimensional plane?

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Problem 33.14

(1)(2)(3)

Write the equations:

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(1)

of the line passing through the points $A(- 1, 0, 3, - 2)$ and $B(2, 1, 4, 5)$

(2)

of the two-dimensional plane passing through the points $A(- 2, 1, 1, 1), B(1, 3, - 5, 2)$ and $C(0, 1, 1, 4)$

(3)

of the three-dimensional plane (hyperplane) passing through the points $A(1, 1, 0, - 1), B(2, - 1, 3, 3), C(1, - 1, 1, 5)$ and $D(0, 0, 3, - 1)$ .

Problem 33.15

Let $A\left(x_{1}^{\prime }, x_{2}^{\prime }, \ldots , x_{n}^{\prime }\right)$ and $B\left(x_{1}^{\prime \prime }, x_{2}^{\prime \prime }, \ldots , x_{n}^{\prime \prime }\right)$ be two distinct points, and let $p$ and $q$ be some numbers. Find the coordinates of the point $C$ dividing the segment $AB$ in the ratio $p: q$ .

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Problem 33.16

Let the points $A, B, C$ in an $n$ -dimensional space not lie on one line. Prove that the medians of the triangle $ABC$ pass through one point and are divided at this point in the ratio $2: 1$ , counting from the vertex.

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Problem 33.17

A point $M$ belongs to the hyperplane given by the equation $a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} = 0$ , and the vector $\overline{MM_{1}}$ has coordinate column $\left(a_{1}, a_{2}, \ldots , a_{n}\right)^{T}$ . Prove that the coordinates of the point $M_{1}$ satisfy the inequality $a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} > 0$ .

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Problem 33.18

Write the parametric equations of the plane given by the system of linear equations:

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(1)

$\mathbf{A}_{27} \mathbf{x} = \mathbf{c}_{46}$ . $\mathbf{A}_{27} = \left[\begin{smallmatrix} 1 & -3 \\ -2 & 6 \end{smallmatrix}\right], \mathbf{c}_{46} = \left[\begin{smallmatrix} 2 \\ -4 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{10} \mathrm{x} = \mathbf{c}_{29}$ . $\mathbf{A}_{10} = \left[\begin{smallmatrix} 2 & 5 \\ 1 & 3 \end{smallmatrix}\right], \mathbf{c}_{29} = \left[\begin{smallmatrix} 3 \\ 2 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{198} \mathbf{x} = \mathbf{c}_{123}$ . $\mathbf{A}_{198} = \left[\begin{smallmatrix} -2 & 8 & 6 \\ -4 & 10 & 6 \\ 4 & -8 & -4 \end{smallmatrix}\right], \mathbf{c}_{123} = \left[\begin{smallmatrix} 12 \\ 12 \\ -8 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{249} \mathrm{x} = \mathbf{c}_{124}$ . $\mathbf{A}_{249} = \left[\begin{smallmatrix} -3 & 1 & -2 \\ 6 & -2 & 4 \\ -15 & 5 & 10 \end{smallmatrix}\right], \mathbf{c}_{124} = \left[\begin{smallmatrix} -3 \\ 6 \\ -15 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{267} \mathbf{x} = \mathbf{c}_{66}$ . $\mathbf{A}_{267} = \left[\begin{smallmatrix} -1 & 3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{517} \mathbf{x} = \mathbf{c}_{125}$ . $\mathbf{A}_{517} = \left[\begin{smallmatrix} 3 & 1 & -2 & 4 \\ 2 & -3 & 6 & -5 \\ 8 & -1 & 2 & 3 \end{smallmatrix}\right], \mathbf{c}_{125} = \left[\begin{smallmatrix} 5 \\ 3 \\ 13 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{403} \mathrm{x} = \mathbf{c}_{208}$ . $\mathbf{A}_{403} = \left[\begin{smallmatrix} 2 & 3 & -1 \\ 4 & 7 & 5 \\ -1 & -2 & -3 \\ 2 & 5 & 13 \end{smallmatrix}\right], \mathbf{c}_{208} = \left[\begin{smallmatrix} -4 \\ -2 \\ -1 \\ 8 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{586} \mathbf{x} = \mathbf{c}_{123}$ . $\mathbf{A}_{586} = \left[\begin{smallmatrix} -3 & 3 & 12 & 6 & -9 \\ -3 & 3 & 12 & 6 & -9 \\ 2 & -2 & -8 & -4 & 6 \end{smallmatrix}\right], \mathbf{c}_{123} = \left[\begin{smallmatrix} 12 \\ 12 \\ -8 \end{smallmatrix}\right]$ .

Problem 33.19

Write a system of equations defining the given plane:

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(1)

$\mathbf{x} = \mathbf{c}_{28} + t\mathbf{c}_{33}$ . $\mathbf{c}_{28} = \left[\begin{smallmatrix} -1 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{33} = \left[\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{x} = \mathbf{c}_{83} + t_{1} \mathbf{c}_{84} + t_{2} \mathbf{c}_{66}$ . $\mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{x} = \mathbf{c}_{147} + t\mathbf{c}_{146}$ . $\mathbf{c}_{147} = \left[\begin{smallmatrix} 5 \\ -2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{146} = \left[\begin{smallmatrix} 1 \\ 4 \\ 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{x} = \mathbf{c}_{168} + t\mathbf{c}_{207}$ . $\mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{207} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

(5)

$\mathbf{x} = \mathbf{c}_{199} + t_{1} \mathbf{c}_{166} + t_{2} \mathbf{c}_{200}$ . $\mathbf{c}_{199} = \left[\begin{smallmatrix} 3 \\ -5 \\ 7 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{200} = \left[\begin{smallmatrix} -1 \\ 8 \\ -6 \\ 5 \end{smallmatrix}\right]$ .

Problem 33.20

Write the equation of the hyperplane in a four-dimensional space passing through the point $M(- 1, 2, 3, 5)$ parallel to the hyperplane $2x_{1} + 3x_{2} - 4x_{3} + x_{4} + 5 = 0$ .

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Problem 33.21

Write the equation of the line in a four-dimensional space passing through the point $M(- 1, 3, 4, 0)$ parallel to the line $x_{1} = 2 + 3t, x_{2} = - 1 + t, x_{3} = 7t, x_{4} = 2 - t$ .

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Problem 33.22

(1)(2)(3)

Write the equations of the three-dimensional plane in a five-dimensional space passing:

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(1)

through the point $M(0, 1, - 1, 3, 4)$ parallel to the three-dimensional plane $x_{1} + 2x_{2} + 3x_{3} = x_{4}, x_{1} + x_{2} + x_{3} + x_{4} = 2x_{5}$

(2)

through the points $M_{1}(1, 3, 1, 0, 1)$ and $M_{2}(0, 0, 1, 1, - 1)$ parallel to the two-dimensional plane $x_{1} + x_{2} - 1 = 0, x_{1} - x_{3} + x_{4} = 0$ , $x_{1} + x_{3} - x_{5} + 1 = 0$

(3)

through the points $M_{1}(- 1, 2, 0, 0, 4), M_{2}(1, 1, 1, 1, 1), M_{3}(0, 1$ , $3, - 1, 1)$ parallel to the line $x_{1} = 1 + 2t, x_{2} = 3 - t, x_{3} = 4, x_{4} = 1 + t, x_{5} = - t$ .

Problem 33.23

(1)(2)

Let $\mathrm{l}$ and $\mathrm{m}$ be two planes in an affine space with directing subspaces $\mathcal{L}$ and $\mathcal{M}$ respectively, passing: $\mathrm{l}$ — through a point $A$ , $\mathrm{~ m}$ — through a point $B$ . Prove that:

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(1)

the intersection $\mathrm{l}\cap \mathrm{m}$ is nonempty if and only if the vector $\overrightarrow {AB}$ belongs to the subspace $\mathcal{L} + \mathcal{M}$

(2)

if the planes $\mathrm{l}$ and $\mathrm{m}$ intersect, then the intersection $\mathrm{l}\cap \mathrm{m}$ is a plane with directing subspace $\mathcal{L} \cap \mathcal{M}$ .

Problem 33.24

Let two planes of dimensions $k_{1}$ and $k_{2}$ in an $n$ -dimensional affine space have a common point, and $k_{1} + k_{2} > n$ . Prove that the dimension of the intersection of the given planes is not less than $k_{1} + k_{2} - n$ . Give formulations of this statement for all possible cases for $n = 3$ and $n = 4$ .

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Problem 33.25

Let a plane $\mathrm{l}$ with directing subspace $\mathcal{L}$ pass through a point $A$ , and a plane $\mathrm{m}$ with directing subspace $\mathcal{M}$ pass through a point $B$ not coinciding with $A$ . Prove that there exists a unique plane of smallest dimension containing both $\mathrm{l}$ and $\mathrm{m}$ ; moreover, the directing subspace of the required plane is the sum $\mathcal{L} + \mathcal{M} + \mathcal{P}$ , where $\mathcal{P}$ is the subspace spanned by the vector $\overrightarrow {AB}$ .

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Problem 33.26

State and prove a statement analogous to the statement in problem 33. 25, for three planes.

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Problem 33.27

(1)(2)(3)

Write the equations of the given plane in a four-dimensional space:

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(1)

of the two-dimensional plane containing the point $A(- 1, 0, 2, 3)$ and the line $x_{1} = 1 - t, x_{2} = 3 + 2t, x_{3} = 1 + t, x_{4} = 3t$

(2)

of the two-dimensional plane containing the parallel lines $x_{1} = - 1 + 2t, x_{2} = t, x_{3} = 0, x_{4} = - 5 - t$ and $x_{1} = 3 + 2t$ , $x_{2} = - 4 + t, x_{3} = 1, x_{4} = - t$

(3)

of the three-dimensional plane containing the point $A(- 3, 0, 1, 0)$ and the two-dimensional plane $x_{1} - x_{2} + x_{3} - 1 = 0, x_{1} + x_{2} + x_{4} = 0$ .

Problem 33.28

(1)(2)(3)

Write the equations of the plane of smallest dimension containing two given planes of a five-dimensional space:

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(1)

the lines $x_{1} = 1 - t, x_{2} = 2 + 3t, x_{3} = 4t, x_{4} = - t, x_{5} = 3$ and $x_{1} = 2 + t, x_{2} = 2t, x_{3} = 1 + t, x_{4} = - 1 + 2t, x_{5} = 3 - t$

(2)

the line $x_{1} = 2 + t, x_{2} = - t, x_{3} = - 1 + t, x_{4} = 1 + 2t$ , $x_{5} = - 3t$ and the two-dimensional plane $x_{1} = t_{1} + 3t_{2}, x_{2} = - 1 + 4t_{1} - - t_{2}, x_{3} = - 3 + t_{1} + t_{2}, x_{4} = 4 - t_{1} + t_{2}, x_{5} = - 2 + t_{2}$

(3)

the two-dimensional planes $x_{1} - x_{3} + x_{4} - 1 = 0, x_{1} + 2x_{4} - - x_{5} - 2 = 0, x_{2} + x_{3} - 2 = 0$ and $x_{1} = x_{2} = x_{3} = 1$ .

Problem 33.29

(1)(2)

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(1)

Prove that if two planes in an $n$ -dimensional space are absolutely skew, then the sum of their dimensions does not exceed $n - 1$ .

(2)

Prove that if two planes in an $n$ -dimensional space are skew parallel to an $r$ -dimensional plane, then the sum of their dimensions does not exceed $n + r - 1$ .

Problem 33.30

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

Investigate the relative position of a line and a two-dimensional plane in a four-dimensional space, if the two-dimensional plane is given by the equations $x_{1} - 2x_{3} + 1 = 0, x_{1} + 2x_{2} - 3x_{3} + x_{4} - 2 = 0$ , and the line is given parametrically:

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(1)

$x_{1} = 3 + 2t, \quad x_{2} = 5, \quad x_{3} = 2 + t, \quad x_{4} = t$

(2)

$x_{1} = - 2 + 3t, \quad x_{2} = 3 - t, \quad x_{3} = - 1 + 2t, \quad x_{4} = - 4 + 4t$

(3)

$x_{1} = 6 + t, \quad x_{2} = 5 - t, \quad x_{3} = 1 + 2t, \quad x_{4} = 1 + 3t$

(4)

$x_{1} = - 1 + 2t, \quad x_{2} = 1 + t, \quad x_{3} = t, \quad x_{4} = 1 - t$ .

Problem 33.31

Investigate the relative position of two two-dimensional planes in a five-dimensional space, if the first plane is given by the equations $x_{1} = x_{2} = 1, x_{3} + x_{4} = x_{5}$ , and the second by parametric equations:

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(1)

$x_{1} = 2 + t_{1}, \quad x_{2} = 3, \quad x_{3} = 3 + 2t_{2}, \quad x_{4} = 4, \quad x_{5} = 5 + t_{1} + t_{2}$

(2)

$x_{1} = - t_{1}, \quad x_{2} = 3 + 2t_{1}, \quad x_{3} = 2 + t_{1}, \quad x_{4} = 1 + t_{1} - t_{2}$ , $x_{5} = 2 + t_{2}$

(3)

$x_{1} = 2 + t_{1} + t_{2}, \quad x_{2} = 3 + t_{1} + t_{2}, \quad x_{3} = 3 + 2t_{1} + t_{2}$ , $x_{4} = 4 + t_{1}, \quad x_{5} = 5 - 2t_{2}$

(4)

$x_{1} = t_{1} - t_{2}, \quad x_{2} = 1, \quad x_{3} = t_{1}, \quad x_{4} = 1 - t_{2}, \quad x_{5} = 3 - - t_{1} + t_{2}$

(5)

$x_{1} = 1, \quad x_{2} = 4, \quad x_{3} = 1 + t_{1} + t_{2}, \quad x_{4} = 2 + 2t_{1} - 2t_{2}$ , $x_{5} = - 5 + 3t_{1} - t_{2}$

(6)

$x_{1} = 1, \quad x_{2} = 1, \quad x_{3} = 2 + 2t_{1} + t_{2}, \quad x_{4} = - 3 + t_{1} - 3t_{2}$ , $x_{5} = - 1 + 3t_{1} - 2t_{2}$ .

Problem 33.32

Prove that two lines in a four-dimensional space, given by the equations $\mathbf{x} = \mathbf{c}_{209} + t\mathbf{c}_{197}$ and $\mathbf{x} = \mathbf{c}_{210} + t\mathbf{c}_{201}$ , have a unique common point. Find the coordinates of this point and write the equations of the two-dimensional plane containing the given lines. $\mathbf{c}_{209} = \left[\begin{smallmatrix} 5 \\ 1 \\ 9 \\ 9 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{210} = \left[\begin{smallmatrix} 1 \\ 5 \\ -2 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{201} = \left[\begin{smallmatrix} 1 \\ -7 \\ 5 \\ -2 \end{smallmatrix}\right]$ .

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Problem 33.33

The points of an affine space are polynomials of degree at most 4, and the vectors are the same polynomials: $\overline{p_{1}(t) p_{2}(t)} = p_{2}(t) - p_{1}(t)$ . The first line contains the points $2t^{4} - 2t$ and $t^{4} + t^{3} - t$ , the second line contains the points $5 + 10t^{2} + 2t^{3}$ and $- 1 - 2t^{2} + 2t^{3}$ . Prove that these lines have a unique common point, and find this point (polynomial).

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Problem 33.34

Write the parametric equations of the line in a four-dimensional space containing the point with coordinate column $\mathbf{c}_{211}$ and intersecting the lines $\mathbf{x} = \mathbf{c}_{212} + t\mathbf{c}_{202}$ and $\mathbf{x} = \mathbf{c}_{213} + t\mathbf{c}_{210}$ ; find the coordinates of the points of intersection. $\mathbf{c}_{211} = \left[\begin{smallmatrix} 1 \\ 4 \\ -1 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{212} = \left[\begin{smallmatrix} 3 \\ 1 \\ 4 \\ 8 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{213} = \left[\begin{smallmatrix} 4 \\ 17 \\ -5 \\ 18 \end{smallmatrix}\right], \mathbf{c}_{210} = \left[\begin{smallmatrix} 1 \\ 5 \\ -2 \\ 5 \end{smallmatrix}\right]$ .

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Problem 33.35

The system of points $A_{1}, \ldots , A_{k}, B_{1}, \ldots , B_{j}$ is independent. Prove that there exist disjoint planes $\mathrm{l}$ and $\mathrm{m}$ of dimensions $k - 1$ and $j - 1$ respectively such that the plane $\mathrm{l}$ contains the points $A_{1}, \ldots , A_{k}$ , and the plane $\mathrm{m}$ contains the points $B_{1}, \ldots , B_{j}$ .

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Problem 33.36

(1)(2)

Let $\mathrm{l}$ and $\mathrm{m}$ be planes of an affine space such that the space of vectors of the affine space is the direct sum of the directing subspaces $\mathcal{L}$ and $\mathcal{M}$ of these planes. Prove that:

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(1)

the projection of any point of the affine space onto the plane $\mathrm{l}$ parallel to the plane $\mathrm{m}$ is uniquely defined

(2)

the projection of any vector $\overrightarrow {AB}$ onto the plane $\mathrm{l}$ parallel to the plane $\mathrm{m}$ is the projection of this vector onto the subspace $\mathcal{L}$ parallel to the subspace $\mathcal{M}$ .

Problem 33.37

(1)(2)

Find the coordinates of the projection of the point $M(5, 0, - 3, 4)$ of a four-dimensional space:

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(1)

onto the hyperplane $x_{1} + x_{2} - x_{3} + 2x_{4} = 2$ parallel to the line $x_{1} = 1 - t, x_{2} = 3 + 4t, x_{3} = 3t, x_{4} = 1 + t$

(2)

onto the two-dimensional plane $x_{1} - x_{2} + x_{3} + 1 = 0, x_{1} + x_{2} = x_{4}$ parallel to the two-dimensional plane $x_{1} + x_{2} + x_{3} + x_{4} = 0, x_{1} - - 2x_{4} - 3 = 0$ .

Problem 33.38

Is the set of points in an $n$ -dimensional affine space $(n = 1, 2, \ldots )$ , whose coordinates $x_{1}, \ldots , x_{n}$ in a Cartesian coordinate system satisfy the condition, convex:

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(1)

$a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} = 0$

(2)

$a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} \geqslant 0$

(3)

$\lambda_{1} x_{1}^{2} + \ldots + \lambda_{n} x_{n}^{2} \leqslant 1$ , where $\lambda_{i} \geqslant 0, \quad i = 1, \ldots , n$

(4)

$\lambda_{1} x_{1}^{2} + \ldots + \lambda_{n} x_{n}^{2} \geqslant 1$ , where $\lambda_{i} \geqslant 0, \quad i = 1, \ldots , n$

(5)

$x_{1} \geqslant 0, x_{2} \geqslant 0, \ldots , x_{n} \geqslant 0$ ;

(6)

$x_{1} x_{2} \ldots x_{n} \geqslant 0$ ?

Problem 33.39

Prove the convexity of a $k$ -dimensional parallelepiped.

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Problem 33.40

Prove that the intersection of convex sets is a convex set.

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Problem 33.41

Find the projection of the four-dimensional simplex bounded by the hyperplanes $x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0$ and $x_{1} + x_{2} + x_{3} + x_{4} = 1$ , onto the hyperplane $x_{1} + x_{2} + x_{3} + x_{4} = 0$ parallel to the line $x_{1} = x_{2} = x_{3} = x_{4}$ .

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Problem 33.42

Prove that all diagonals of a parallelepiped intersect at one point, called the center of the parallelepiped.

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Problem 33.43

(1)(2)

For a $k$ -dimensional parallelepiped, find the number of:

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(1)

distinct $p$ -dimensional faces

(2)

distinct diagonals.

Problem 33.44

Determine the shape and vertices of the cross-sections of the four-dimensional parallelepiped $- 1 \leqslant x_{i} \leqslant 1, i = 1, 2, 3, 4$ , by the hyperplane $x_{1} + x_{2} + x_{3} + x_{4} = 0$ .

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