Linear Algebra: Polynomials subspaces

In summary, U and W are subspaces of V = P3(R) and given the subspace U{a(t+1)^2 + b | a,b in R} and W={a+bt+(a+b)t^2+(a-b)t^3 |a,b in R} there is a basis for U perp for some inner product. The solution for finding U perp is valid if the vectors are linearly independent.
  • #1
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U and W are subspaces of V = P3(R)
Given the subspace U{a(t+1)^2 + b | a,b in R} and W={a+bt+(a+b)t^2+(a-b)t^3 |a,b in R}

1) show that V = U direct sum with W
2) Find a basis for U perp for some inner product

Attempt at the solution:

1) For the direct sum I need to show that it satisfy two conditions (a) V = U + W (which I did), (b) the intersection of U and W = {0} which I am not quite sure how to do.

2) expanding U: a(t^2 + 2t +1) +b. as far as I understand U perp needs only to contain t^3 so that U + U perp span P3 (right?).

Next I used the standard inner product for polynomials such that the integral from 0 to 1 of U.U perp = 0.

so I have the following (at^2 + 2at +a +b)(-bt^3) integrate from 0 to 1, set it equal to zero and solve for b in terms of a. (here I chose U perp to be -bt^3 from some b in R).

Is solution valid for finding a basis for U perp? Thanks
 
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  • #2
For the b part of 1, you might try this.
U = {a(t + 1)2 + b | a, b in R}
W = {(c - d)t3 + (c + d)t2 + dt + c | c, d in R}
(Notice that I changed to c and d in the 2nd subspace.)

You can split these up like so:
U = { a(t2 + 2t + 1) + b(1)}
W = {c(t3 + t2 + 1) + d(-t3 + t2 + t)}

If you think about the generators of each subspace as vectors (which are probably a bit easier to work with), U is all linear combinations of <0, 1, 2, 1> and <0, 0, 0, 1>, and W is all linear combinations of <1, 1, 0, 1> and <-1, 1, 1, 0>.

If you can show that these vectors are linearly independent, that means that none of them is a linear combination of the others, and that means that the intersection of the two subspaces is {0}.
 
  • #3
Thanks Mark44 for your help
 

1. What is a polynomial subspace?

A polynomial subspace is a subset of a vector space that consists of all possible linear combinations of polynomials. This means that any polynomial in the subspace can be expressed as a sum of other polynomials in the subspace, with coefficients from the underlying field.

2. How are polynomial subspaces related to linear algebra?

Polynomial subspaces are a fundamental concept in linear algebra, as they involve the study of vector spaces and their properties. In particular, polynomial subspaces are used to understand the structure of vector spaces and the behavior of linear transformations.

3. What are the basis and dimension of a polynomial subspace?

The basis of a polynomial subspace is a set of polynomials that span the subspace, meaning that any polynomial in the subspace can be written as a linear combination of the basis polynomials. The dimension of a polynomial subspace is the number of basis polynomials, and it represents the number of independent directions within the subspace.

4. How can polynomial subspaces be represented geometrically?

Polynomial subspaces can be represented geometrically as a set of vectors in a higher-dimensional space. The basis vectors of the subspace can be thought of as axes in this space, and any polynomial in the subspace can be represented as a linear combination of these basis vectors.

5. How are polynomial subspaces used in real-world applications?

Polynomial subspaces have many real-world applications, such as in computer graphics, cryptography, and signal processing. They are also used in statistics and data analysis for creating polynomial models to fit data and make predictions. Furthermore, polynomial subspaces are widely used in engineering and physics for solving differential equations and representing physical systems.

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