Expressing a Linear Transformation With Respect to a Non Standard Basis

Ifis a polynomial of degree 2, then the transformationis a linear transformation. We can express this transformation in matrix form with respect to the spandard basis inby considering the effect of g on each of the basis elements

With respect to the standard basis, this transformation may be represented by the matrix

Suppose we wanted to represent this transfomation with respect to a non standard basis, sayWe must consider the effect of the transformation on each of the basis elementsand express the resiult in terms of these basis elements to obtain the new transformation matrix.

We need to express these results in terms of the new basis elements.

We obtain the three vectorshence the transformation matrix is

The transformation matrix from the new basis to the old basis is(this expresses the new basis vectors in terms of the old ones). The connection between the original transfomation matrix A and the new one B is

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