Linear spaces

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M

- be a set of polynomials

p \in P_{n}

with real coefficients satisfying the specified conditions. Prove that

M

- is a linear subspace in

P_{n}

, find its basis and dimension. Complement the basis of

M

to the basis of the entire space

P_{n}

. Find the transition matrix from the canonical basis of the space

P_{n}

to the constructed basis.

n = 3, M = {p \in P_{3} ∣ p^{''} (a) + p^{'} (0) = 0} .

1.9.1 Linear spaces

3.85 $

Problem: Prove that the set of matrices

M

is a subspace in the space of all matrices of a given size. Construct a basis and find the dimension of the subspace

M

. Check that the matrix

B

M

and decompose it according to the found basis.

M = {A \in M_{3 \times 3} ∣ A = A^{T}

(symmetrical), the sums of the elements in the columns are the same, sums of elements in rows alternate\},

B = (\begin{array}{ccc} 0 & 1 & - 1 \\ 1 & - 1 & 0 \\ - 1 & 0 & 1 \end{array})

1.9.2 Linear spaces

6.42 $

Problem: Find out, does the set of all real numbers forms the linear space if the sum of any two elements

a

b

is defined in it, equal to

a + b

and the product of any element

a

by any real number

ε

ε \cdot a

1.9.3 Linear spaces

0 $

Problem: Prove that the set

M

x (t)

, given on the area D, forms a linear space. Find its dimension and basis.

M = {α + β \tan t + γ \cot t}, t \in (0, \frac{π}{2}) .

1.9.4 Linear spaces

2.57 $

V

- linear space of all symmetric polynomials of degree at most two over

R

from two variables

x

y

. Choose a basis in space

V

and find the operator matrix

L

in this basis, if

L (f) (x, y) = (2 x + 3 y) \frac{\partial f}{\partial x} + (3 x + 2 y) \frac{\partial f}{\partial y} .

1.9.5 Linear spaces

3.85 $

Problem: Prove that the set of vectors

L = {\bar{a} = (α_{1}, α_{2}, \dots, α_{n}) ∣ α_{1} + α_{2} + \dots + α_{n} = 0}

is the subspace of the space

R^{n}

, a sequence of

n

-dimensional vectors

{\bar{a}}_{1} = (1, 0, \dots, 0, - 1)

{\bar{a}}_{2} = (0, 1, \dots, 0, - 1), \dots, {\bar{a}}_{n - 1} = (0, 0, \dots, 1, - 1)

basis of this subspace.

1.9.6 Linear spaces

2.57 $

Problem: Prove that the set of

n

-dimensional vectors

L = {\bar{a} = \underset{n}{\underset{⏟}{(α, β, α, β, \dots)}} ∣ α, β \in R}

is the subspace of the space

R^{n}

, find the basis and dimension of this subspace.

1.9.7 Linear spaces

2.57 $

M

be a set of polynomials

P \in P_{n}

with real coefficients satisfying the specified conditions. Prove that

M

- linear subspace in

P_{n}

, find its basis and dimension. Complement basis

M

to the basis of the whole space

P_{n}

n = 3, M = {P \in P_{3} ∣ P^{''} (1) + P^{'} (0) = 0} .

1.9.8 Linear spaces

3.08 $