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

In summary, to find a basis for the set S of polynomials of degree less than or equal to 4 where f(0) = f(1) = 0, we can use the system of equations a + b + c + d = 0 and e = 0 to find a solution set of <-1, 1, 0, 0, 0>, <-1, 0, 1, 0, 0>, <-1, 0, 0, 1, 0>, giving us a basis of {-x4 + x3, -x4 + x2, -x4 + x} or {x4 - x3, x4 - x2,
  • #1
Buri
273
0

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
 
Last edited:
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  • #2
Buri said:

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.
Buri said:
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}.
 

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

What is a subspace?

A subspace is a subset of a vector space that still holds all the properties of a vector space, such as closure under addition and scalar multiplication.

How is the degree of a subspace determined?

The degree of a subspace is determined by the highest degree of any polynomial in the subspace. In this case, the degree of the subspace S is ≤ 4.

What does f(0)=f(1)=0 mean in this context?

This means that the subspace S contains polynomials that have a value of 0 when evaluated at both 0 and 1. In other words, the subspace contains polynomials that have a root at both 0 and 1.

How do you find the basis for a subspace?

To find the basis for a subspace, you need to find a set of linearly independent vectors that span the subspace. In this case, we are looking for a set of polynomials that have a degree ≤ 4 and have roots at 0 and 1.

Why is finding the basis for a subspace important?

Finding the basis for a subspace allows us to represent any vector in that subspace as a linear combination of the basis vectors. This can help with solving systems of linear equations and understanding the structure of the subspace.

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