Linear spaces

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Problem: Let \( M \) - 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. \[ n=3, M=\left\{p \in P_{3} \mid p^{\prime \prime}(a)+p^{\prime}(0)=0\right\} \text {. } \]

1.9.1 Linear spaces

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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 \( \mathrm{B} \) belongs to \( \mathrm{M} \) and decompose it according to the found basis. \( M=\left\{A \in M_{3 \times 3} \mid A=A^{T}\right. \) (symmetrical), the sums of the elements in the columns are the same, sums of elements in rows alternate\}, \[ B=\left(\begin{array}{ccc} 0 & 1 & -1 \\ 1 & -1 & 0 \\ -1 & 0 & 1 \end{array}\right) \]

1.9.2 Linear spaces

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Problem: Find out, does the set of all real numbers forms the linear space if the sum of any two elements \( a \) and \( b \) is defined in it, equal to \( a+b \) and the product of any element \( a \) by any real number \( \varepsilon \), equal to \( \varepsilon \cdot a \).

1.9.3 Linear spaces

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Problem: Prove that the set \( M \) of functions \( x(t) \), given on the area D, forms a linear space. Find its dimension and basis. \[ M=\{\alpha+\beta \tan t+\gamma \cot t\}, \quad t \in\left(0, \frac{\pi}{2}\right) . \]

1.9.4 Linear spaces

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Problem: Let \( V \) - linear space of all symmetric polynomials of degree at most two over \( \mathbb{R} \) from two variables \( x \) and \( y \). 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

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Problem: Prove that the set of vectors \( L=\left\{\bar{a}=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \mid \alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}=0\right\} \) is the subspace of the space \( R^{n} \), a sequence of \( n \)-dimensional vectors \( \bar{a}_{1}=(1,0, \ldots, 0,-1) \), \( \bar{a}_{2}=(0,1, \ldots, 0,-1), \ldots, \bar{a}_{n-1}=(0,0, \ldots, 1,-1) \quad \) basis of this subspace.

1.9.6 Linear spaces

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Problem: Prove that the set of \( n \)-dimensional vectors \( L=\{\bar{a}=\underbrace{(\alpha, \beta, \alpha, \beta, \ldots)}_{n} \mid \alpha, \beta \in R\} \quad \) is the subspace of the space \( \mathbb{R}^{n} \), find the basis and dimension of this subspace.

1.9.7 Linear spaces

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Problem: Let \( M \) be a set of polynomials \( P \in \mathrm{P}_{n} \) with real coefficients satisfying the specified conditions. Prove that \( M \) - linear subspace in \( \mathrm{P}_{n} \), find its basis and dimension. Complement basis \( M \) to the basis of the whole space \( P_{n} \). \[ n=3, \quad M=\left\{P \in \mathrm{P}_{3} \mid P^{\prime \prime}(1)+P^{\prime}(0)=0\right\} . \]

1.9.8 Linear spaces

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