[SOLVED] MA502 Homework 3

1. Given two vector basis B1 = {v₁,…,v_n} and B2 = {w₁,…,w_n} in a vector space V and a linear transformation L : V → V , prove that

[L]_B2_→B1[a]_B2 = [B2 ^→B1][L]_B1_→B2[a]_B1

for any a ∈ V . (Hint: show separately that each side is identical to

[L(a)]_B1.)

2. Consider the linear map L : R³ →R³ represented in canonical coordinates by the matrix

Find (1) The Null space; (2) The Range. Determine if the linear systems

Lv = (1,2,0)

Lv = (6,8,6)

have a solution, if it is unique or not. If there exists at least a solution compute one.

3. Consider the operator T(p) = ^R p(x)dx from the space of all polynomials P to itself. Compute its Null space and its range. (Note: P is not a finite dimensional space)
4. Is it possible for a linear map from R³ → R¹⁰⁰ to be onto? Explain your answer in detail.
5. Is it possible for a linear map from R¹⁰⁰ →R³ to be one to one? Explain your answer in detail. Is it possible for such a map to be onto? If your answer is yes do provide an example.