Find Basis for Subspace: S with Degree ≤ 4 & f(0)=f(1)=0

[Buri]

Homework Statement

I need to find a basis for the following:

S = {f are polynomials of degree less than or equal to 4| f(0) = f(1) = 0}

2. The attempt at a solution

A general polymial is of the form:

p(x) = ax^4 + bx^3 + cx^2 + dx + e

Now for p(0) = p(1) = 0 I must have:

e = 0 and a + b + c + d + e = 0

Which basically becomes:

a + b + c + d = 0.

Here is where I get stuck. I'd usually solve for a variable and then substitute it back into the polynomial and then group like coefficients to get a basis, but it just isn't working!

Any help?

NEVERMIND! I've figured out what I was doing wrong lol Apparently I don't know how to factor LOL

 

[Mark44]

Homework Statement

I need to find a basis for the following:

S = {f are polynomials of degree less than or equal to 4| f(0) = f(1) = 0}

2. The attempt at a solution

A general polymial is of the form:

p(x) = ax^4 + bx^3 + cx^2 + dx + e

Now for p(0) = p(1) = 0 I must have:

e = 0 and a + b + c + d + e = 0

Which basically becomes:

a + b + c + d = 0.

And e is still 0.

Here is where I get stuck. I'd usually solve for a variable and then substitute it back into the polynomial and then group like coefficients to get a basis, but it just isn't working!

Any help?

NEVERMIND! I've figured out what I was doing wrong lol Apparently I don't know how to factor LOL

Notwithstanding that you have figured this out, you can find the solution set for the system of equations
a + b + c + d = 0
e = 0

by simply solving for a.

a = -b - c - d
b = b
c = ...c
d = ...d
e = 0

Then any "vector" of coefficients looks like <a, b, c, d, e> = b<-1, 1, 0, 0, 0> + c<-1, 0, 1, 0, 0> + d<-1, 0, 0, 1, 0>, where b, c, and d are any real scalars.

Putting this back in terms of polynomials, a basis is {-x⁴ + x³, -x⁴ + x², -x⁴ + x}.

Another basis, with all leading coefficients positive is {x⁴ - x³, x⁴ - x², x⁴ - x}.