Prove that every inner product on R^2 is generated by some 2x2 matrix

[cgossman]

Prove that every inner product on R^2 is generated by some 2x2 matrix. I'm pretty bad at doing proofs or working with general forms all together. any help would be much appreciated.


Thanks!

 

[HallsofIvy]

A mapping from \(\displaystyle R^2\) to R can be written as \(\displaystyle \begin{pmatrix}x & y\end{pmatrix}\begin{pmatrix} a & b \\ c & d\end{pmatrix}\begin{pmatrix}u \\ v\end{pmatrix}\). See what happens to the various inner products of the basis vectors.

 

[cgossman]

hm. I don't quite see the connection between that and the inner product. could you provide an example?