0
$\begingroup$

Question:

Let $P_{3}(\mathbb{R})$ have the standard inner product and $U$ be the subset spanned by the two vectors (which are polynomials) $u_{1}=1+2x-3x^2$ and $u_{2}=x-x^2+2x^3$. Find the basis for the orthogonal complement $U^{⊥}$.

I honestly have no idea how to approach this question. I know what orthogonal complement and a basis are but I don't understand where to begin or even solve this question. Any help would be much appreciated. Thanks in advance.

$\endgroup$
  • 1
    $\begingroup$ What is the inner product? I am not sure I am aware of a standard one for polynomials of a fixed $\max$ degree. $\endgroup$ – copper.hat Mar 26 at 3:51
  • $\begingroup$ It is a good question. I have assumed that $$\langle f(x),g(x)\rangle = \int_{0}^{1}f(x)g(x)\mathrm{d}x$$ $\endgroup$ – user1337 Mar 26 at 3:59
0
$\begingroup$

Here is a laborious way:

You know $\dim P_3 = 4$ and one basis is $e_k(x) = x^k$, $k=0,...,3$.

Now apply Gram Schmidt to the collection $u_1,u_2,e_1,e_2,e_3,e_4$ (discarding any elements if the Gram Schmidt process reduces it to zero). Suppose the result is $v_1,v_2,v_3,v_4$, then $v_3,v_4$ will span $U^\bot$.

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

HINT

Every finite dimensional inner product vector space $V$ admits the decomposition $V = U\oplus U^{T}$, where $U$ is a linear subspace of $V$.

At your case, $V = P_{3}(\textbf{R})$ and $U = \text{span}\{1 + 2x - 3x^{2}, x - x^{2} + 2x^{3}\}$.

In order to find the basis for the orthogonal complement $U^{T}$, consider the inner product defined on $V$ according to \begin{align*} \langle f(x),g(x)\rangle = \int_{0}^{1}f(x)g(x)\mathrm{d}x \end{align*}

Since $\dim W_{1} + \dim W_{2} = \dim(W_{1}+W_{2}) + \dim(W_{1}\cap W_{2})$, we conclude that \begin{align*} \dim V = \dim(U\oplus U^{T}) & = \dim U + \dim U^{T} - \dim(U\cap U^{T}) = \dim U + \dim U^{T} \end{align*} Given that $\dim V = 4$ and $\dim U = 2$, the basis of $U^{T}$ consists of two vectors. With the purpose of finding $u_{3}$ and $u_{4}$ that spans $U^{T}$, it suffices to solve the system of equations obtained from \begin{align*} \langle u_{1},u\rangle = \langle u_{2},u\rangle = 0 \end{align*} where $u\in P_{3}(\textbf{R})$. Can you take it from here?

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.