Description

 1. Consider the vector space F

2
2
. What are all the possible bases for this space? (You do
not need to prove this).
2. Consider the linearly independent set in R
3 and construct a basis by adding one element
to it. Then prove that it is a basis.
{(1, 0, 1),(−1, 0, 1)}
3. Consider the linearly independent set in R
3 and construct a basis by adding one element
to it. Then prove that it is a basis.
{(1, −1, 1),(0, 1, 1)}
4. In the previous two examples, you wrote two distinct bases of R
3
. Given a vector
(x, y, z) ∈ R
3
, write a set of functions fi(a1, a2, a3) = bi where each function takes in
the coefficients of the first basis and produces the i
th coefficient of the second basis.
What kind of functions are these?
5. Prove or give a counterexample: If v1, v2, v3, v4 is a basis of V and U is a subspace of
V such that v1, v2 ∈ U and v3 ̸∈ U and v4 ̸∈ U, then v1, v2 is a basis of U.
1