Given two vector basis B1 = {v1,...,vn} and B2 = {w1,...,wn} 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.
