Inner product of matrix relative to a base

Question

find the matrix $A$ that represents the usual inner product on $R^2$ relative to the given base of $R^2$ $S=\{(1,3),(2,5)\}$. How do I proceed if the same question would be for an inner product on $R^3$ and again, a suitable basis. What is the logic? and how is this matrix related to the positive definite matrix?

Answer 1

compute $a_{11}=<v_1,v_1>$, $a_{12}=<v_1,v_2>=<v_2,v_1>=a_{21}$ $,a_{22}=<v_2,v_2>$.Then your desired matrix will be $\begin{bmatrix} a_{11}& a_{12}\\ a_{21}&a_{22} \end{bmatrix} $