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Fuli Name: Group:

3.02.03

Linear Algebra Examination

Task l(7p.)

a)    Using the Euclidean algorithm, express the GCD(21, 31) as the sum 21<z + 31&, a,beZ.

b)    Find an integer d suchthal d = 2(mod21) and d = -4(mod 31). Does a positive number satisfying both of the conditions exist?

Task 2(7p.)

Solve the following eąuation:

|z|² +(1 + t)z = 0, zeC

Task 3(6p.)

Let A =

-1	r	, B =	4-1 2'
0	2		3 5 -3

. If it is possible calculate 2A^rA + 3B-B^r.

Task 4(8p.)

Find the dependence of the number of Solutions of the following system of eąuations on the parameter a:

2x + ay+2z = a + 2 • x+y+z=a 4 x + (a- l)y + 2z = 8

Task 5(8p.)

a)    Solve the following system of eąuations using the Gauss method:

z, + 2x₂ - 3*3 + x_Ą + 2x₅ = 0 < x, + 4x₂ - 5Xj + 4x₅ = 0 x, +2x₂ -3x₃ +x₄ +3x₅ = 0

b)    Find the basis and the dimension of the solution space.

c)    Find the coordinates of the vector (-3,2,1,2, 0) relative to the determined basis.

Task 6(7p.)

The matrix of the linear transformation f:R³^>R² relative to the unit basis has the

following form:

A =

1 2 -3 4 5    3*

a)    Find the formula which defines /.

b)    Find the basis for the image of /.

Task 7(7p.)

Let f:R³-+R³, f(x,y,z) = (-2x-3y + z,3y + z,3z) be a linear transformation. Find the eigenvalues of / and an eigenvector associated with one of the eigenvalues.