# Thread: Probably an easy question: finding a transformation R^3 -> R^3

1. ## Probably an easy question: finding a transformation R^3 -> R^3

Hello,

My overall goal is to find a linear transformation which takes R^3 -> R^3, taking the inner product space with squared norm x^2 + 2xy + 4y^2 + 8z^2 into the std. inner product.

Now, I know how to use the gram-schmidt method to orthonormalize a basis, but I am just completely blanking on how to find a basis to orthonormalize! Any ideas would be great. I don't need the problem solved, just the right nudge

Thanks!

2. I've done some more work on the problem and have reduced it further. I believe this is a notational problem on my end, as I have two books for the class and the notation is inconsistent.

I am given that the "squared norm" is as given above. How do I extract from that the inner product?

I assume for a vector v = (x, y, z), ||v||^2 = x^2 + 2xy + 4y^2 + 8z^2 = <v, v>, but how would I define this for different vectors <v, u>?

I hope I'm being clear. Thanks for the help!

3. Originally Posted by arcketer
I assume for a vector v = (x, y, z), ||v||^2 = x^2 + 2xy + 4y^2 + 8z^2 = <v, v>, but how would I define this for different vectors <v, u>?
Write the expression for the norm as $\|v\|^2 = (x+y)^2+3y^2+8z^2.$ Then the inner product will be given by $\langle v,u\rangle = (x_1+y_1)(x_2+y_2) + 3y_1y_2 + 8z_1z_2$, where $v=(x_1,y_1,z_1)$ and $u=(x_2,y_2,z_2).$

4. Oh wow, thank you very much. I'm embarassed for not having noticed that haha.