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Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: Let M - be a set of polynomials p∈Pn with real coefficients satisfying the specified conditions. Prove that M - is a linear subspace in Pn, find its basis and dimension. Complement the basis of M to the basis of the entire space Pn. Find the transition matrix from the canonical basis of the space Pn to the constructed basis. n=3,M={p∈P3∣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 belongs to M and decompose it according to the found basis. M={A∈M3×3∣A=AT (symmetrical), the sums of the elements in the columns are the same, sums of elements in rows alternate\}, B=(01−11−10−101)

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 and b is defined in it, equal to a+b and the product of any element a by any real number ε, equal to ε⋅a.

1.9.3 Linear spaces

0 $

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={α+βtan⁡t+γcot⁡t},t∈(0,π2).

1.9.4 Linear spaces

2.57 $

Problem: Let V - linear space of all symmetric polynomials of degree at most two over 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)=(2x+3y)∂f∂x+(3x+2y)∂f∂y.

1.9.5 Linear spaces

3.85 $

Problem: Prove that the set of vectors L={a¯=(α1,α2,…,αn)∣α1+α2+⋯+αn=0} is the subspace of the space Rn, a sequence of n-dimensional vectors a¯1=(1,0,…,0,−1), a¯2=(0,1,…,0,−1),…,a¯n−1=(0,0,…,1,−1) basis of this subspace.

1.9.6 Linear spaces

2.57 $

Problem: Prove that the set of n-dimensional vectors L={a¯=(α,β,α,β,…)⏟n∣α,β∈R} is the subspace of the space Rn, find the basis and dimension of this subspace.

1.9.7 Linear spaces

2.57 $

Problem: Let M be a set of polynomials P∈Pn with real coefficients satisfying the specified conditions. Prove that M - linear subspace in Pn, find its basis and dimension. Complement basis M to the basis of the whole space Pn. n=3,M={P∈P3∣P′′(1)+P′(0)=0}.

1.9.8 Linear spaces

3.08 $

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