Polynomials : Matrices, Reflection Matrices and Linear Transformation

From the cubics P3 to the fourth degree polynomials P4, what matrix represents multiplication by 2 + 3t? The columns of the 5 by 4 matrix A come from applying the transformation to each basis vector x1 = 1, x2 = t, x3 = t2, x4 = t3.

Verify directly from c2 + s2 = 1 that the reflection matrices satisfy H2 = I.
c = cosine
s = sine

Prove that A2 is a linear transformation if A is (say from R3 to R3).

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Solution Summary

Matrices, Reflection Matrices and Linear Transformation are investigated. The solution is detailed and well presented.