this question is for proof based linear algebra please prove the following

Problem 3. Let V and W be vector spaces of dimensions a and m, respectively, and let T : V > H be a linear transformation. (a) Prove that for every pair of ordered bases B = ('51, . mm) of V and C = [131, . ..,1m) of W, ther exists a unique m x a matrix A such that [Tfik = Ale for all a? E V. The matrix A is called th (Em-matrix of T, written A = c[T]3. [b] For each n E N, let P\" be the vector space of polynomials of degree at most a in the variable t. Le D : P3 > \"P2 be the differentiation map. Let s=(1,r,s,s), seesaw), C=(1,t,t2], C'=(1,2t,3t2), so that B, B\" are ordered bases of \"Pa, and C, C\" are ordered bases of \"Pg. Find C[D] and c:[D]3:. [c] Prove that there exist ordered bases 3 = [if], . .. ,n) of V and C = (131, . .. ,fm) of W such the c[T]3 has block form clTls = [%|%] v where k = rank') and the remaining blocks are appropriately sized zero matricesdr [d] Prove that for any or x a matrix A, there exist an invertible m x m matrix P and an invertible H Xi matrix Q such that FAQ has block form = Wt where k = ranh[A) and the remaining blocks are appropriately sized zero matrims