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## Main Question or Discussion Point

Just started working through "Linear Algebra Done Right". There is something I don't understand.

Given b ∈ F, then

{(x

is a subspace of F

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?

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=0I 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?