B 04.02.05

Linear Algebra Examination

Name and Surname:

Group number:

Note: Please write each solution on a separate sheet of paper.

1. Let z₁ = 1+2i, z₂ = -2-i. Find
.

2. Find the real and imaginary parts of the solution of the following equation

3. a *) Find the basis and the dimension of the linear space

b) Let

Determine whether W is a linear subspace of R⁴. If so find the basis of W.

4 Use the Kronecker-Capelli theorem to determine the dependence of the number of solutions of the following system on the value of the parameter p.

.

5. Solve the following system of equations

6. Find a value of the parameter a, for which the set of vectors (v1, v2, v3), where v1=(1,1,1), v2=(2,3,1), v3=(5,a,1) constitutes a basis of R³. the coordinates of the vector (0,1,0) relative to this basis.

7. Let the linear transformation F be given by the formula
. Find the matrix of the transformation, and the bases of Ker F and Im F.

8. Let the linear transformation be given by
,
. Find the

eigenvalues and eigenvectors of F and the dimensions of linear spaces associated with these vectors. Do they constitute a basis of R³ . Is so write the matrix A_F relative to this basis.

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