B

16.01.06

ALGEBRA TEST 2

Full Name and Group number..................................................................

REMEMBER TO JUSTIFY ALL YOUR ANSWERS!!!

1. Use two different methods to find the inverse
of the following matrix

.

2. Determine the relationship between the value of the parameter 'p' and the number of solutions of the following system of equations.

3. Find the determinant of

.

4. Use the definition to prove or disprove that the following subsets
, are linear subspaces of
:

,

.

5. Find the values of
for which the vectors v₁ = (2, a), v₂ = (4, 3) constitute a basis of
. Choose one of these values of a and find the coordinates of v relative to the determined basis. The coordinates of the vector v relative to the unit basis are (2,3).

6. Let
be a linear transformation
. Determine the matrix of
, the bases and dimensions of its kernel Ker
and image Im
.

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