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Consider the function $d/dx$ from $P_n$ (the real vector space of degree $\leq n$ polynomials in one variable $x$) to $P_{n-1}$.

a) Prove that $d/dx$ is a linear transformation.

b) Write the associated matrix of the linear transformation with respect to the standard basis $(\{1,x,x^2,\dots,\})$ for $P_n$ and $P_{n-1}$.

c) What is the kernel of $d/dx$? What is range of $d/dx$?

I proved that $d/dx$ is a linear transformation by showing that it holds under additivity and homogeneity. However, I'm stuck on b). I tried to align the standard basis with its derivatives however I am confused as to what to do for the derivatives of $1$ and $x$.

I know that the kernel of $d/dx$ is all constants and the range is going to be $P_{n-1}$.

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Well, what is the derivative of $1$? What is the derivative of $x$? – Chris Eagle Oct 30 '12 at 23:46
0 and 1 respectively – tamefoxes Oct 31 '12 at 1:13
up vote 1 down vote accepted

The action of a linear transformation is completely determined by their action on a basis of the space. If we apply $\frac{\rm d}{\rm dx}$ to the standard basis, then $$\frac{\rm d}{\rm dx}x^k = kx^{k-1}$$ for each $k\in\{0,\ 1,\ \cdots,\ n\}$. What space do these vectors span? What dimension is it? What dimension is the kernel then?

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The vectors span P_(n-1). The dimension of it is n-1. The dimension of the kernel is 1. – tamefoxes Oct 31 '12 at 1:14
Right, this shows that the range is simply $P_{n-1}$. You know that the constants are part of the kernel and that the kernel is dimension $1$. This shows that the constants span the kernel. – EuYu Oct 31 '12 at 1:16
Thanks. For the second part of the question, is the associated matrix a diagonal matrix where the entries starting with 1 as the first diagonal entry, and increasing by n+1? – tamefoxes Oct 31 '12 at 1:27
Seems like it should start with zero. – EuYu Oct 31 '12 at 1:32
Sweet, thanks a lot! – tamefoxes Oct 31 '12 at 1:34

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