Find the matrix representation of the derivative map ddx : P3(R) → P3(R), withrespect to the basis {1, x, x2, x3}
Question
Find the matrix representation of the derivative map ddx : P3(R) → P3(R), withrespect to the basis {1, x, x2, x3}
Solution
The derivative map d/dx : P3(R) → P3(R) is a linear transformation that takes a polynomial p(x) in P3(R) and maps it to its derivative p'(x).
The basis for P3(R) is given as {1, x, x^2, x^3}.
Let's find the matrix representation of the derivative map with respect to this basis.
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Apply the derivative map to each basis vector:
d/dx(1) = 0 d/dx(x) = 1 d/dx(x^2) = 2x d/dx(x^3) = 3x^2
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Express each result as a linear combination of the basis vectors:
0 = 01 + 0x + 0x^2 + 0x^3 1 = 11 + 0x + 0x^2 + 0x^3 2x = 01 + 2x + 0x^2 + 0x^3 3x^2 = 01 + 0x + 3x^2 + 0x^3
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The coefficients of these linear combinations give the columns of the matrix representation:
The matrix representation of the derivative map d/dx with respect to the basis {1, x, x^2, x^3} is:
| 0 1 0 0 | | 0 0 2 0 | | 0 0 0 3 | | 0 0 0 0 |
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