A 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 = 2+3i, z₂ = -3 -2i. 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=(1,2,3), v2= (3,2,1), v3= (4,a,5) constitutes a basis of R³. Find the coordinates of the vector (1,0,0) 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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