Question 1
Let Λ
3(R
n) be the vector space of all 3-forms on R
n and Λ
5(R
n) be the vector space of all 5-forms on R
n. If Λ
3(R
n) and Λ
5(R
n) have the same dimension, find n.
◦ 4
◦ 8 or -1
◦ 15
◦ 2
◦ 8
Question 2
Let η = F
1dy∧dz - F
2dx∧dz+ F
3 dx∧dy ∈Λ
2(R
3) and let
u= a
1 i+ a
2 j+ a
3 k and
v = b
1 i+ b
2 j+ b
3 k
be vectors in R
3. If η is identified by the vector
w= F
1 i+ F
2 j+ F
3 k then
η (
u,
v) is equal to
◦
w ∙ (
v × u)◦
w × (
u × v)◦
w ∙ (
u + v)◦
w ∙ (
u × v)◦
w ∙ (
v - u)
