# Orthogonal Basis

Let $\mathbb{R^2}$ have the inner product definied by the positive definite matrix

$K=\pmatrix{2&-1\\-1&3}$

a.) Show that $v_1 = (1,1)^T$, $v_2 = (-2,1)^T$ form an orthogonal basis.

b.) Write the vector v = $(3,2)^T$ as a linear combination of $v_1,v_2$ also find an orthonormal basis $u_1,u_2$ for this inner product and write v as a linear combination of the orthonormal basis.

You do not have to necessarily answer the question I just need help setting it up. What I think should happen for b is just Ax = b so

$Ax=\pmatrix{1&-2\\1&1}$ and b = $K=\pmatrix{-2\\1}$ and just row reduce and it will give me the linear combinations and to for the second part of b I will have to convert the matrix to be orthogonal and then find the normal vector. For part a, I do not know what to do?

• I'm glad to know I don't have to answer the question. You had me worried there for a moment. Oct 25 '12 at 6:26
• Anyway, do you know what it means to speak of the inner product defined by some given matrix? Do you know what it means for two vectors to be orthogonal with respect to some given inner product? Oct 25 '12 at 6:28
• For the first part, not necessarily and for the second part of your question I know in order to be orthogonal the dot product needs to be equal to zero, but it asks for the inner product (like you said) so do I have to convert the matrix into a quadratic function? Oct 25 '12 at 6:35
• Do you not have a textbook or some lecture notes to tell you what it means to speak of the inner product defined by a given matrix? Oct 25 '12 at 12:22

An inner product defined with respect to a positive definite matrix is given by $$\langle v_1 , v_2 \rangle = v_1^T A v_2$$ To show that the two vectors are orthogonal, show that the above inner product is zero for the vectors $v_1$ and $v_2$.
For part (b), try to write $v$ as $\alpha v_1 + \beta v_2$. (Find $\alpha$ and $\beta).$ For the second half of the part (b), get orthonormal vectors by dividing $v_1$ and $v_2$ by their respective norms induced by the inner product.
• To get an orthonormal basis, you do exactly what Marvis wrote: you divide $v_1$ and $v_2$ by their norms. Oct 25 '12 at 12:20