B 30.01.07

Linear Algebra Examination

Full Name:

Group number:

Task 1 (14p.) Show that
is a root of the following polynomial:
.

Find all the other roots .

Task 2 (18p.) Find the basis for
,

.

Does the vector
belong to S? (that is:
)

Write the coordinates of v with respect to the above basis.

Task 3 (16p.) Use the Gaussian Elimination method to solve the following system of equations:

Task 4 (20p.) Find the real and imaginary parts Re z, Im z and plot the values of z for
.

Task 5 (16p.) The linear transformation
is defined by

.

Find the eigenvalues and eigenvectors of this transformation. Do they constitute a basis

for
? Can the matrix of L be written in diagonal form? If so, write this matrix.

Task 6 (16p.) Determine the relationship between the value of the parameter 'q' and the number of

solutions of the following system of equations.

---