
Inner Products
Hey everyone I am not sure how to solve this problem:
Let <u,v> be the inner product on $\displaystyle \mathbb{R}^2$ generated by
$\displaystyle \begin{bmatrix}2&1 \\ 1&1 \end{bmatrix}$
and let u = (2,1), v=(1,1), w=(0,1)
I need to compute <u,v>
Can someone help me with this?
Thanks

The entries of the matrix tell you what the inner product does to basis elements, e.g. the top lefthand element of the matrix tells you what you get when you do <e_1 , e_1>. If you know what it does to the basis elements you can extend it to any vector.

I'm a little confused, how would I would use the entries in the matrix to find the inner product?

Let {i,j} be the standard basis of R^2. Then the matrix of the inner product gives you <i,i>,<i,j>,<j,i>,<j,j> going from left to right, then down. Can we use these numbers to work out what the inner product of any two vectors in R^2? (Hint: write the vectors in terms of their basis elements)

Can you give an example of this type of problem, if possible?
Thanks

Let x = ai + bj, y=ci+dj in R^2
Then <x,y> = <ai+bj,ci+dj> = <ai + bj,ci> + <ai+bj,dj> = <ai,ci> + <bj,ci> + <ai,dj> + <bj,dj> = ac<i,i> + bc<j,i> + ad<i,j> + bd<j,j>. Then you can use the matrix. If you don't understand any of those steps I suggest you look up the definition of a real inner product:
Inner Product  from Wolfram MathWorld