# Show that derivation is linear?

• Mar 23rd 2011, 12:12 PM
posix_memalign
Show that derivation is linear?
I'm completely stuck on this one:

"Derivation $Dp = p'$ defines a transformation $D : P_2 \rightarrow P_2$. Use the definition of linear maps and the properties of derivation to show that D is linear. Describe the codomain and the range of D. What is the kernel of D? What are the dimension of range(D) and kernel(D)? Is the map onto? Is it one-to-one? Is it an isomorphism?"

Where should I start? What should I read in order to understand this? Is this a problem that is understood simply through general experience with calculus and linear algebra?
• Mar 23rd 2011, 12:32 PM
Drexel28
Quote:

Originally Posted by posix_memalign
I'm completely stuck on this one:

"Derivation $Dp = p'$ defines a transformation $D : P_2 \rightarrow P_2$. Use the definition of linear maps and the properties of derivation to show that D is linear. Describe the codomain and the range of D. What is the kernel of D? What are the dimension of range(D) and kernel(D)? Is the map onto? Is it one-to-one? Is it an isomorphism?"

Where should I start? What should I read in order to understand this? Is this a problem that is understood simply through general experience with calculus and linear algebra?

Which part don't you understand? the fact that the derivative defines a linear transformation is a trivial fact from calculus. So, for the kernel, you tell me? In general one of the guys in $\mathcal{P}_2$ looks like $a+bx+cx^2$ and so we want to solve this general guy for $0(x)=D(a+bx+cx^2)=b+cx$. Recall then that a polynomial is zero, by definition, if and only if it has all zero coefficients Lastly, what do you think the image is? Let me ask you, for example, does there exists $p\in\mathcal{P}_2$ such that $p'(x)=x^2$?
• Mar 23rd 2011, 12:34 PM
TheEmptySet
Quote:

Originally Posted by posix_memalign
I'm completely stuck on this one:

"Derivation $Dp = p'$ defines a transformation $D : P_2 \rightarrow P_2$. Use the definition of linear maps and the properties of derivation to show that D is linear. Describe the codomain and the range of D. What is the kernel of D? What are the dimension of range(D) and kernel(D)? Is the map onto? Is it one-to-one? Is it an isomorphism?"

Where should I start? What should I read in order to understand this? Is this a problem that is understood simply through general experience with calculus and linear algebra?

First note that a basis for

$\mathbb{P}_2=\{1,x,x^2\}=\{\mathbf{e}_1,\mathbf{e} _2,\mathbf{e}_3\}$

Now using the definition of the derivative we get that

$T(\mathbf{e}_1)=D(1)=0$
$T(\mathbf{e}_2)=D(x)=1=\mathbf{e}_1$
$T(\mathbf{e}_3)=D(x^2)=2x=2\mathbf{e}_2$

So this transformation has the matrix representation

$\begin{bmatrix}0 & 1& 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{bmatrix}$

Does this help? see if you can finish from here!