Algebra-linear transformations and vectors

Question

Questions:

Algebra: linear transformations and vectors

1 Prove that the solution space of AX = 0, where A is a m x n matrix, is avector space.

2 Are the vectors x³ - 1, x² - x and x linearly independent in P₃ ? Why ?

3 Determine whether or not the function T : M^mn→M^mn defined by T(A) = A + B, where B is a mixed m x n matrix, is a linear transformation. If it is a linear transformation, verify this fact.

4 The function T : R² → R² such that T[(x; y)] = (-x, y) is called a reflection in the y-axis. Is this function a linear transformation ?

5 Find the kernel and the range for the linear transformation T : R²→R² given by T[(x; y)] = (2x; x - y)

6 Let the linear transformation T : Mⁿⁿ→ Mⁿⁿ be defined by T(A) = A+A^t.
Find ker(T).

7 Find the change-of-basis matrix from B to B' where
                            B = {(3,-2),(6, 8)} and B' = {(1, 0); (0, 1)}

8 Let the linear transformation T : R² → R² be defined by T[(x,y)] = (2x + y, x - y).

                     Find [T]_BB where B = {(2,-3),(4,5)} and B' = {e₁; e₂}

9 Find the dimension of the solution space of the following homogenous system of linear equations;
                             x + y - 3x = 0
                            4x + y + 5z = 0
                             2x + y + 6z = 0

10 Find the rank and nullity of the linear transformation T : R² → R²given by T(u) = Proj_v = u where v = (2,-4).

11 Find the 3x3 matrix that describes the following mapping in R2; scaling by 6 in the x-direction and by -8 in the y-direction.

12 Prove that A is similar to A for every n x n matrix A.