Polynomial Span and Subspace - Linear Algebra

[1LastTry]

Homework Statement

Consider the vector space F(R) = {f | f : R → R}, with the standard operations.
Recall that the zero of F(R) is the function that has the value 0 for all
x ∈ R:
Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have
the same value at x = −1 and x = 1.
Define functions g; h; j and k ∈ F[R] by
g(x) = 2x3 − x − 2x2 + 1; h(x) = x3 + x2 − x + 1;
k(x) = −x3 + 5x2 + x + 1 and j(x) = x3 − x; ∀x ∈ R:
a) Show that g and h belong to U.
b) Show that k ∈ span{g; h}.
c) Show that j =∈ span{g; h}.
d) Show that span{g; h} ̸= span{g; h; j}.

Homework Equations

I don't really know any equation relevant, I do not really understand the concept behind Polynomial spans.

I am unsure about this:

is the span of a polynomial...lets say x^2+5x-3 and x^2+3x+10 is the span just span{x^2,x,-4, and x^2,x,10}? I think you can see where I tried to attempt at this question (B), since both are x^3, x^2, x type of thing I said it was in the same span and etc.

My method (probably wrong) worked for (C), however I got stuck on (D), where
I am not sure what the span means.

The Attempt at a Solution

 

[Robert1986]

Homework Statement

Consider the vector space F(R) = {f | f : R → R}, with the standard operations.
Recall that the zero of F(R) is the function that has the value 0 for all
x ∈ R:
Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have
the same value at x = −1 and x = 1.
Define functions g; h; j and k ∈ F[R] by
g(x) = 2x3 − x − 2x2 + 1; h(x) = x3 + x2 − x + 1;
k(x) = −x3 + 5x2 + x + 1 and j(x) = x3 − x; ∀x ∈ R:
a) Show that g and h belong to U.
b) Show that k ∈ span{g; h}.
c) Show that j =∈ span{g; h}.
d) Show that span{g; h} ̸= span{g; h; j}.

Homework Equations

I don't really know any equation relevant, I do not really understand the concept behind Polynomial spans.

I am unsure about this:

is the span of a polynomial...lets say x^2+5x-3 and x^2+3x+10 is the span just span{x^2,x}?


I think you can see where I tried to attempt at this question (B), since both are x^3, x^2, x type of thing I said it was in the same span and etc.

My method (probably wrong) worked for (C), however I got stuck on (D), where
I am not sure what the span means.

The Attempt at a Solution

One thing that might help you when considering these questions about polynomials, is that you can think of a polynomial as being a vector. For example, x^2 + x + 1 can be viewed as the vector [1,1,1] and the polynomial 3x^2 + 5 can be viewed as the vector [3,0,5]. In fact, the set of polynomials of degree less than or equal to n is pretty much the same as the R^(n-1). (In math, we have a fancy term for this:isomorphic.)

 

[1LastTry]

One thing that might help you when considering these questions about polynomials, is that you can think of a polynomial as being a vector. For example, x^2 + x + 1 can be viewed as the vector [1,1,1] and the polynomial 3x^2 + 5 can be viewed as the vector [3,0,5]. In fact, the set of polynomials of degree less than or equal to n is pretty much the same as the R^(n-1). (In math, we have a fancy term for this:isomorphic.)

so uhhh... do you think the spans like a vector too? For example x^2+x+1 which you said it is like [1,1,1] so is the span [1,0,0] + [0,1,0] + [0,0,1]? I am wondering can you show some steps you would do for a) or b). You don't have to do the problem for me, give me some hints on how to tackle this question...


Thanks