Determine transformation matrixf:P2(R)→P2(R) is given by:f(p(x))=3⋅p(1)−x2⋅p(0)+(x−1)⋅p′(1)

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missygouldzl · Accepted Answer

Step 1  A linear transformation can always be represented as a matrix, once you have chosen a basis for the domain and codomain. Since f is a map from a three-dimensional vector space to itself, it has a representation as a 3×3 matrix:  [f(p)1f(p)2f(p)3]=[?????????][p1p2p3]  where the argument to f is p(x)=p1+p2x+p3x2 and the result is f(p(x))=f(p)1+f(p)2x+f(p)3x2  Now can you &quot;probe&quot; the ?s by plugging in different polynomials into f? For example, if you plug in p(x)=1,  which in the monomial basis is the vector  [100]  what is f(p), in the monomial basis? What does this tell you about the entries of the matrix? How do you find the rest of the ?s

trnovitom06 · Accepted Answer

Find f(1), f(x) and f(x2). Take the coefficients of 1, x and x2 in f(1) and turn it into a column. For example   f(1)=3−x2  so the first column of the transformation matrix  (30−1)  Do the same for f(x) and f(x2) to get the second and third column of transformation matrix.

Determine transformation matrix f: P_2(\mathbb{R}) \to P_2(\mathbb{R}) is given by: f(p(x))=3 \cdot

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