Linear Transformations: Polynomial to its second derivatives

[Javé]

Consider the Vector Space of Polynomial with degree <= 5.
And the transformation to its second derivatives.
Shows the transformation and its matrix em relate to the base of this space.


Could anybody help me?
Thanks!

 

[Steven G]

Sure someone could help you! Can you please tell us what help you need? Tell us where you are stuck? Show us the work you have done so far. If you give us nothing to work with then it is hard to help you solve your problem.

 

[HallsofIvy]

First, a linear transformation on a vector space can be written as a matrix in a given basis. What is the "given basis" for this problem? The "standard basis" for the vector space of polynomials of degree less than or equal to 5 is {1, x, x^2, x^3, x^4, x^5}. Is that what is intended?

Second, to get that matrix representation, you apply the linear transformation to each basis vector, in turn, and write the result as a linear combination of the basis vectors. That gives each column of the matrix. For example the second derivative of both 1 and x is 0 which is just 0 times each basis vector. The first two columns of the matrix are all 0s. The second derivative of x^2 is 2= 2(1)+ 0(x)+ 0(x^2)+ 0(x^3)+ 0(x^4)+ 0(x^5) so the third column is $\displaystyle \begin{bmatrix}2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}$. The second derivative of x^3 is 6x so the fourth column is $\displaystyle \begin{bmatrix}0 \\ 6 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}$. The second derivative of x^4 is 12x^2 so the fifth column is $\displaystyle \begin{bmatrix}0 \\ 0 \\ 12 \\ 0 \\ 0 \\ 0\end{bmatrix}$. The second derivative of x^5 is 20x^3 so the sixth column is $\displaystyle \begin{bmatrix}0 \\ 0 \\ 0 \\ 20 \\ 0 \\ 0\end{bmatrix}$.

The matrix representation of this linear transformation, in the standard basis is $\displaystyle \begin{bmatrix} 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 12 & 0 \\ 0 & 0 & 0 & 0 & 0 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$.