2
$\begingroup$

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?

  • 0
    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?2012-10-25

1 Answers 1

2

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.

  • 0
    To get an orthonormal basis, you do exactly what Marvis wrote: you divide $v_1$ and $v_2$ by their norms.2012-10-25