Instructions.

1. Formulate the theorem about the rank and the nullity of a matrix. Use this theorem to prove that if a subspace W of Rⁿ has dimension m then the orthogonal complement W^c has dimension n-m.

2. Is it true that the map from C(0,1) to R² which takes every function f(x) from C(0,1) to the vector (f(1)+f(0)², f(1)+f(0)) is a linear transformation?

3. What is the standard matrix of the rotation of R² through the angle Pi/4?

4. Represent the following matrix as a product of elementary matrices.

                              [-7   4]
                              [-4   2]

5. (i) For which values of a is the following matrix A invertible.
(ii) In the case when it is invertible, find the inverse.
(iii) What is the rank of this matrix depending on a?
(iv) For which values of a the linear operator in R³ with standard matrix A is surjective?
(v) What is the dimension of the null space of this matrix depending on a?
(vi) What is the basis of the column space of this matrix depending on a?

                              [a-1 1   1  ]
                      A =  [0    a   1  ]
                              [0    0  a-2]

6. For which values of parameter a do the following vectors form a basis in R²: (a², 2a), (1,1).

7. For which values of k is the matrix

        [1 -2]
        [k  0]

        [ 3  0]           [ 2 -1]
        [-2  0],          [-5  0]

8. Find the determinant of the following n by n matrix:

                      [1 1 1 1.... 1]
                      [0 1 1 1.... 1]
                      [0 0 1 1.....1]  
                      ....................
                      [1 0 0 0 ....1]

9. (i) Find the orthogonal basis in the subspace of R⁴ spanned by the following 3 vectors: (1,1,1,1), (1,-1,1,1), (1,1,-1,1).
(ii) Find the projection of the vector (1,0,0,0) onto this subspace.

10. Let T be the linear operator in R² with the following standard matrix:

                [ 3 2]
                [-1 0]