Let the matrix A = (aij) represent a linear operator with respect to an orthonormal basis x1;

Question:

Let the matrix A = (aij) represent a linear operator with respect to an orthonormal basis x1; x2,..., xn for an inner product space X. Then
aij = xTf (xj) for every i, j
A link between the inner product and the familiar geometry of ℜ3 is established in the following exercise, which shows that the inner product is a measure of the angle between two vectors.