What is the basis and dimension of the subspace U of P2?

In summary, the basis and dimension of the subspace U of P2 where p(1) = p(2) is a(x^2 - 3x) + c, with a dimension of 2. The notation P2 refers to the space of all second degree polynomials.
  • #1
Hockeystar
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Homework Statement


Find the basis and dimension of the following subspace U of P2

p(x) [tex]\ni[/tex] P2 such that p(1) = p(2)

Homework Equations


The Attempt at a Solution



I know all quadratics are in the form ax2 + bx + c

set p(2) = p(1)

4a + 2b + c = a + b + c
b = -3a

Therefore ax2 -3a + c

Basis(U) = a(x2-3x) + c

Therefore dim(U) = 2

I'm just wondering if I have the correct answer or not. Going into linear algebra midterm tomorrow and prof never really went over polynomials but it's on the test.
 
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  • #2
I assume P2 is the space of all second degree polynomials (the notation P2 is not standard, so includi it next time :smile: ). Then your proof is correcT.

The basis for U is by the way [tex] \{x^2-3x,1\} [/tex]. What you wrote down doesn't make much sense to me...
 
  • #3
Thanks for the reply. I just wanted clarification that I'm properly solving the problem. My format was off because I used subscript instead of superscript.
 

1. What is the basis of P2?

The basis of P2, also known as the basis of a vector space, is a set of vectors that are linearly independent and span the entire vector space.

2. How many vectors are needed to form a basis for P2?

In P2, which is a vector space of polynomials of degree 2 or less, a basis can be formed using 3 vectors. These vectors must be linearly independent, meaning that none of them can be written as a linear combination of the others.

3. What is the dimension of P2?

The dimension of P2 is the number of vectors in its basis, which in this case is 3. This means that any vector in P2 can be written as a unique combination of 3 linearly independent vectors.

4. Can a subset of P2 be a basis for the entire vector space?

Yes, a subset of P2 can be a basis for the entire vector space as long as it meets the criteria of being linearly independent and spanning P2. However, the smallest possible basis for P2 is still 3 vectors.

5. How is the basis of P2 related to its dimension?

The basis of P2 and its dimension are directly related - the basis is a set of vectors that has the same number of elements as the dimension of P2. The basis is essentially a representation of the dimension in terms of vectors.

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