For the following maps, determine whether they are linear transformations or not, and present an
appropriate proof.

(a) T : R^4 → M2,3 given by T(a, b, c, d) =  [a   a^2   a^3
                                                                   b    c       d ]
.
(b) T : M2,3 → M3,2 given by T(M) = M^T (transpose of M)
.
(c) T : P3 → P3 given by T(p(x)) = p(2) + 3x · p'(x), where p'(x) denotes the derivative of the polynomial p(x).

i know that the 2 rules for proving are T (u+v)= T(u) + T (v) and T (ku)= k T (u).....but how do i show it with the questions above, like do i just take any numbers , so confused

 V = {(2a + b + c, 3a + b + 2c, 2a + c, b + 7c)|a, b, c ∈ R} forms a subspace of R^4 
 
 Show that {(2, 3, 2, 0),(1, 1, 0, 1),(0, 1, 1, 6)} is an (ordered) basis B for V . 

toally confused on how to do this??..please help!!

let A be a matrix=

[  7        7      9    -17

   6        6      1    -2

 -12    -12    -27    1

   7       7      17   -15 ]

What is the reduced row echelon form of A?

What is the rank of A?

A consistent system of linear equations in 14 unknowns is reduced to row echelon form. There are then 10 non-zero rows (i.e. 10 pivots). How many parameters (free variables) will occur in the solution?