# Homework Help: Ordinary matrix-vector multiplication

1. Mar 5, 2006

Let $$P=(p_{ij})$$ be a real symmetric 2x2 matrix. Show that the function on $$\mathbb{R}^2\times\mathbb{R}^2$$ (Where R^2 is a space of column vectors) defined by $$<v,w>=v^tPw$$ is an inner product if and only if $$p_{11}$$ and $$det(P)$$ are both swtrictly positive.

I just need to know what $$Pw$$ means in $$<v,w>=v^tPw$$.

2. Mar 5, 2006

### arildno

Ordinary matrix-vector multiplication.

3. Mar 5, 2006

I see. I thought it was "P of w".

4. Mar 5, 2006

This function doesn't send vectors to scalars, it can't be an inner product, unless I understood something wrong.

5. Mar 5, 2006

v and w are 1x2 column vectors, right? So why is this function on R^2xR^2? Isn't it defined on the vector space R^2?

6. Mar 5, 2006

### shmoe

they're 2x1, but there are two of them. This function takes a pair of vectors, (v,w) and gives a real number, so the domain is R^2xR^2.

7. Mar 5, 2006

### HallsofIvy

v is in R2, w is in R2 so (v, w) is in R2 x R2.