- 120

- 15

Given b ∈ F, then

{(x

_{1},x

_{2},x

_{3},x

_{4}) ∈ F

^{4}: x

_{3}= 5x

_{4}+ b}

is a subspace of F

^{4}*if and only if* b=0

I just flat out don't understand why b has to be 0 or even what is the point of bringing this up.

and right below that is:

{p ∈ P(F) : p(3) = 0}

is a subspace of P(F).

P(F) refers to the polynomial space. F is the set of fields and it contains C (complex numbers) and R (real numbers).

Again, what is the point of bringing this up and how do we know that p is a subspace of P(F) based off of the information given?