(a) Prove that for any linear map h: V W and any W, the set h-1() has the form { + | N (h)} for

(a) Prove that for any linear map h: V †’ W and any ˆˆ W, the set h-1() has the form
{ + | ˆˆ N (h)}
for ˆˆ V with h() = (if h is not onto then this set may be empty). Such a set is a coset of N (h) and we denote it as + N (h).
(b) Consider the map t: R2 †’ R2 given by

for some scalars a, b, c, and d. Prove that t is linear.
(c) Conclude from the prior two items that for any linear system of the form
ax + by = e
cx + dy = f
we can write the solution set (the vectors are members of R2)
{ + | satisfies the associated homogeneous system}
where is a particular solution of that linear system (if there is no particular solution then the above set is empty).
(d) Show that this map h: Rn †’ Rm 0 is linear

for any scalars a1,1, . . . , am, n. Extend the conclusion made in the prior item.
(e) Show that the k-th derivative map is a linear transformation of Pn for each k. Prove that this map is a linear transformation of the space

for any scalars ck, . . . , c0. Draw a conclusion as above.

x (axby cx dy CX tTL, k- dxk