Hey everyone I am not sure how to solve this problem:
Let <u,v> be the inner product on generated by
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 left-hand 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.
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)
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