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Q. Let A be a real invertible n * n matrix. Show that [tex]\left\langle {\mathop x\limits^ \to ,\mathop y\limits^ \to } \right\rangle \equiv \mathop y\limits^ \to A^T A\mathop x\limits^ \to = \left( {A\mathop y\limits^ \to } \right)^T \left( {A\mathop x\limits^ \to } \right)[/tex] defines an inner product in R^n, where x and y are column vectors in R^n. What happens when A is not invertible? (Note: M^T is the transpose of a matrix M, obtained by intechanging the rows and columns of M).

The first step would be to show that the inner product is symmetric I would say. I think I should get to [tex]... = \mathop x\limits^ \to A^T A\mathop y\limits^ \to [/tex] but I don't know how to do get to it. Can someone suggest a method to use? I'm not sure if I need to explicity write down a matrix in this question.