Math Problems Vault

Problem 993

Prove that for any vector $u$ the equalities $0 \cdot u = 0,(-1) \cdot u = -u$ hold.

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Problem 994

In which of the following cases are the indicated operations on the set $X$ well-defined and do they endow it with the structure of a linear space over the field $\mathbb {k}$ :

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(1)

$\mathbb {k} = \mathbb {R}, X$ is the half-plane $y \geqslant 0$ , the operations of addition and multiplication by scalars are standard (i.e., coordinatewise);

(2)

$\mathbb {k} = \mathbb {R}, X$ is the set of vectors in the plane issuing from the origin whose endpoints lie on a given line; the operations are standard;

(3)

$\mathbb {k} = \mathbb {R}, X$ is the set of vectors in the plane issuing from the origin whose endpoints lie on a given line; the operations of addition « $\widehat{+}$ » and multiplication by scalars « $\widehat{\cdot }$ » are given by the formulas

u \widehat{+} v = u+v-x_{0}, \quad \lambda \widehat{\cdot } u = \lambda u+(1-\lambda ) x_{0}

where $x_{0} \in X$ is a fixed vector;

(4)

$\mathbb {k} = \mathbb {R}, X$ is the set of plane vectors lying in the first and third coordinate quadrants $(x y \geqslant 0)$ ; the operations are standard;

(5)

$\mathbb {k} = \mathbb {R}, X$ is all vectors of space except vectors parallel to some fixed line; the operations are standard;

(6)

$\mathbb {k} = \mathbb {R}, X = \mathbb {Q}$ ; the operations are standard;

(7)

$\mathbb {k} = \mathbb {R}, X = \mathbb {C}$ ; the operations are standard;

(8)

$\mathbb {k} = \mathbb {R}, X = (0, \infty )$ ; the operations of addition « $\widehat{+}$ » and multiplication by scalars « $\widehat{\cdot }$ » are given by the formulas

u \widehat{+} v = u v, \quad \lambda \widehat{\cdot } u = u^{\lambda }

(9)

$\mathbb {k} = \mathbb {Q}, X = (0, \infty ) \cap \mathbb {Q}$ ; the operations are the same as in the previous item;

(10)

$\mathbb {k} = \mathbb {R}, X = \mathbb {R}^{2}$ ; the operations are given as follows:

\left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right) = \left(x_{1}+x_{2}, y_{1}+y_{2}\right), \quad \lambda \cdot (x, y) = (\lambda x, 0)

(11)

$\mathbb {k}$ is arbitrary, $X$ is the set of polynomials over $\mathbb {k}$ of degree exactly $n$ ; the operations are standard;

(12)

$\mathbb {k}$ is arbitrary, $X$ is the set of polynomials over $\mathbb {k}$ of degree at most $n$ ;

(13)

$\mathbb {k} = \mathbb {R}, X$ is the set of monotonic functions on a segment; the operations are standard;

(14)

$\mathbb {k} = \mathbb {R}, X$ is the set of all infinite arithmetic progressions with the standard operations;

(15)

$\mathbb {k} = \mathbb {R}, X$ is the set of all infinite geometric progressions with the standard operations?

Problem 995

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

Can the structure of a linear space be defined:

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(1)

on a set consisting of two elements;

(2)

on the abelian group $\mathbb {Z}$ of integers;

(3)

on the line $\mathbb {R}$ with the following operation of multiplication by scalars: $\lambda \cdot u = \lambda^{2} u$

(4)

on the line $\mathbb {R}$ with the operation of multiplication by scalars $\lambda \cdot u = \lambda^{3} u$ ?

Problem 996

Prove that in a linearly independent system of vectors, every subsystem is also linearly independent.

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Problem 997

Let the system of vectors $a_{1}, \ldots , a_{k}$ be linearly independent, and let the system $a_{1}, \ldots , a_{k}, a_{k+1}$ be linearly dependent. Prove that the vector $a_{k+1}$ is linearly expressed in terms of the others, and that this expression is unique.

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Problem 998

Let the vector $x$ be linearly expressed in two different ways in terms of the vectors $a_{1}, \ldots , a_{k}$ . Prove that the system $a_{1}, \ldots , a_{k}$ is linearly dependent.

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Problem 999

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

Let $a, b, c$ be a linearly independent system of vectors. Will the following systems of vectors be linearly dependent:

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(1)

$a+b+c, a+b, a$ ;

(2)

$a+b, b+c, c+a$ ;

(3)

$a-b, b-c, c-a$

(4)

$\alpha a-\beta b, \gamma b-\alpha c, \beta c-\gamma a$ ?

Problem 1000

If every vector of the system of vectors $y_{1}, \ldots , y_{n}$ is a linear combination of the vectors $x_{1}, \ldots , x_{m}$ , one says that the system $y_{1}, \ldots , y_{n}$ is linearly expressed in terms of the system $x_{1}, \ldots , x_{m}$ . Prove the transitivity of this notion: if the system of vectors $y_{1}, \ldots , y_{n}$ is linearly expressed in terms of the system $x_{1}, \ldots , x_{m}$ , and the system $z_{1}, \ldots , z_{p}$ is linearly expressed in terms of the system $y_{1}, \ldots , y_{n}$ , then the system of vectors $z_{1}, \ldots , z_{p}$ is linearly expressed in terms of the system $x_{1}, \ldots , x_{m}$ .

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Problem 1001

Prove that if $\mathbb {R}$ is considered as a linear space over $\mathbb {Q}$ , then the vectors 1 and $\xi$ from $\mathbb {R}$ are linearly independent if and only if $\xi$ is irrational.

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Problem 1002

Under the conditions of the previous problem, prove that for any $n$ the vectors $1, \xi , \xi^{2}, \ldots , \xi^{n}$ are linearly independent if and only if $\xi$ is a transcendental number.

Problem 1001: $\mathbb {R}$ is considered as a linear space over $\mathbb {Q}$ , $\xi \in \mathbb {R}$ .

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Problem 1003

What conditions must the scalar $\xi$ satisfy for the vectors $(\xi , 1,0),(1, \xi , 1)$ and $(0,1, \xi )$ from $\mathbb {R}^{3}$ to be linearly dependent? What is the answer to the same question if $\mathbb {R}^{3}$ is replaced by $\mathbb {Q}^{3}$ ?

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Problem 1004

Prove the linear independence of all geometric progressions starting with one, in the vector space of infinite sequences.

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Problem 1005

Investigate the linear dependence of the following systems of functions $(n > 0)$ :

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(1)

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

(2)

$\sin x, \sin 2 x, \ldots , \sin n x$

(3)

1, $a_{1} \cos x+b_{1} \sin x, a_{2} \cos 2 x+b_{2} \sin 2 x, \ldots , a_{n} \cos n x+b_{n} \sin n x$ (for each $1 \leqslant i \leqslant n$ the numbers $a_{i}$ and $b_{i}$ do not vanish simultaneously);

(4)

$1, \cos x, \cos^{2} x, \ldots , \cos^{n} x ;$

(5)

$\sin x, \sin^{2} x, \ldots , \sin^{n} x$ ;

(6)

$1, \sin x, \cos x, \sin^{2} x, \cos^{2} x, \ldots , \sin^{n} x, \cos^{n} x$ , where $n > 1$ ;

(7)

$1, \ln x, \ln 2 x, \ldots , \ln n x$ ;

(8)

$2^{\alpha_{1} x}, 2^{\alpha_{2} x}, \ldots , 2^{\alpha_{n} x}$ , where all the numbers $\alpha_{1}, \ldots , \alpha_{n}$ are distinct;

(9)

$x^{\alpha_{1}}, x^{\alpha_{2}}, \ldots , x^{\alpha_{n}}$ , where all the numbers $\alpha_{1}, \ldots , \alpha_{n}$ are distinct.

Problem 1006

Prove that in the space of polynomials, every finite system consisting of polynomials of distinct degrees and not containing zero is linearly independent.

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Problem 1007

What conditions must the scalars $\xi , \eta , \zeta$ satisfy for the vectors $\left(1, \xi , \xi^{2}\right),\left(1, \eta , \eta^{2}\right)$ and $\left(1, \zeta , \zeta^{2}\right)$ to be linearly dependent? Generalize this problem to the case of arbitrary dimension.

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Problem 1008

Prove that in every system $X$ of vectors of the space $\mathbb {k}^{n}$ containing at least one nonzero vector, one can choose a linearly independent subsystem $X^{\prime }$ such that the linear spans $\langle X\rangle$ and $\left\langle X^{\prime }\right\rangle$ coincide. Every such subsystem $X^{\prime }$ is called a base of the given system $X$ .

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Problem 1009

Prove that all bases of a system of vectors consist of the same number of vectors. This number is called the rank of the system.

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Problem 1010

(1)(2)

Prove that in the space $\mathbb {k}^{n}$ :

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(1)

any nonzero vector of a given system can be included in a base of this system;

(2)

any linearly independent subsystem can be extended to a base of this system.

Problem 1011

Prove that if two systems of vectors have the same rank and one of these systems is linearly expressed in terms of the other, then the linear spans of these systems coincide.

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Problem 1012

(1)(2)

Find the rank of the system of vectors:

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(1)

$x_{1} = (1,-1,0,0)$ , $x_{3} = (0,0,1,-1)$ , $x_{2} = (0,1,-1,0), \quad x_{4} = (-1,0,0,1)$ ;

(2)

$x_{1} = (1,1,1,1,1), \quad x_{3} = (1,-1,1,-1,1)$ , $x_{2} = (1, i,-1,-i, 1), \quad x_{4} = (1,-i,-1, i, 1)$ .

Problem 1013

(1)(2)(3)

For each of the following systems of vectors, find some base:

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(1)

$x_{1} = (-1,4,-3,-2)$ ,

$x_{3} = (3,-2,1,0)$ ,

$x_{2} = (3,-7,5,3)$ ,

$x_{4} = (-4,1,0,1)$ ;

(2)

$x_{1} = (14,-27,-49,113)$ ,

$x_{3} = (-29,55,96,-227)$ ,

$x_{2} = (43,-82,-145,340), \quad x_{4} = (85,-163,-292,679) ;$

(3)

$x_{1} = (3-i, 1-2 i,-7+5 i, 4+3 i), x_{2} = (1+3 i, 1+i,-6-7 i, 4 i)$ , $x_{3} = (0,1,1,-3)$ .

Problem 1014

(1)(2)

Find all bases of the following systems of vectors:

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(1)

$x_{1} = (1,2,3,0,-1), \quad x_{2} = (0,1,1,1,0), \quad x_{3} = (1,3,4,1,-1)$ ;

(2)

$x_{1} = (1+i, 1-i, 2+3 i), \quad x_{2} = (i, 1,2)$ , $x_{3} = (1-i,-1-i,-2+3 i), x_{4} = (4,-4 i, 12 i)$ .