MA502 -Homework 3
1. 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 : R
3 → R
3
represented in canonical coordinates by the matrix
[L]C→C =
1 2 3
4 0 4
2 1 3
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 R
3 → R
100 to be onto? Explain
your answer in detail.
5. Is it possible for a linear map from R
100 → R
3
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.
1
