# Represent Linear Transformations by Matrices

I'm having problems with linear transformations of polynomials.

Note: all examples are made up so maybe they aren't very specific, but I'm trying to solve specific doubts, I hope you understand. Basically, I am trying to understand what base should I use to represent a linear transformation in matrix form if the linear transformation involves a polynomial.

I want to represent a linear transformation: for example: T(ax²+bx +c)= 2ax + b If I want to represent it with a matrix I do this:

T(x²)= T(1x²+0x+0) = 2x, T(x) = T(0x²+1x+0) = 1 , T(1) = T(0x²+0x +1) = 0

And then I have to write it as a linear combination of the basis {x²,x,1} (((I'm not really sure about this, I mean if it doesn't give me a specific basis, should I use that?, and not {x,1}))

T(x²) = 2x = 0x² + 2x + 0 , T(x) = 1 = 0x² + 0x + 1 , T(1) = 0 = 0x² + 0x + 0 Then I can create a matrix where: (0,2,0) is column 1 (0,0,1) is column 2 (0,0,0) is column 3 And that would be the final answer. But I keep having doubts with this: If I have T(x,y) = (3x+2y,2y,3x) for example (Assume it's a linear transformation), T(1,0) = (3,0,3) T(0,1) = (2,2,0) Then, T(1,0) = (3,0,3) = 3(1,0,0) + 0 (0,1,0) + 3(0,0,1) T(0,1) = (2,2,0) = 2(1,0,0) + 2 (0,1,0) + 0(0,0,1) After that: (3,0,3) is the first column and (2,2,0) the second column. Is that the formal way to do it? Also, I wrote it like a linear combination of the basis {(1,0,0),(0,1,0), (0,0,1)}. I don't know if you can understand my question. In this exercise it goes from R2-> R3 and I use the base of R3 for the linear combination. In the polynomials exercise, should I use the base that the codomain has? for example: R³[x] -> R²[X] (Here I use basis {x²,x,1} for the linear combination) R³[x] -> R³[X] ( Here I use the basis {x³,x²,x,1) Is that right?

Edit: Please assume linear transformations to be true. I made them up in order to clear up a specific doubt I had.

• Everything you wrote is correct. Commented Dec 15, 2022 at 14:08
• You should stick to a concrete recipie . For a linear transformation $T:V\to W$ . such that $\dim(V)=n$ and $\dim(W)=m$.You first fix bases $\{v_{1},...,v_{n}\}$ for $V$ and $\{w_{1},...,w_{m}\}$ for $W$. Now $T(v_{1})$ lies in $W$ and hence can be expressed as a linear combination of the basis vectors. So let $T(v_{1})=\sum_{i=1}^{m}c_{i1}w_{i1}$. Then $(c_{1},...,c_{m})^{T}$ would be the first column of the $m\times n$ matrix of $T$. In this way , you do for all general $v_{i}$. And you get $T(v_{i})=\sum_{j=1}^{m}c_{ji}w_{j}$ . Thus you get the ij th element of the matrix is $c_{ij}$ Commented Dec 15, 2022 at 14:15
• Now sometimes you will be given the bases wrt which you have to compute the matrix. So you'll need to follow the recipie above. Otherwise if nothing is given. Then you can work with any pair of bases of your choice. Most times you'll want the standard ordered bases for your spaces. Commented Dec 15, 2022 at 14:18

## 1 Answer

Represent a polynomial by its coefficients as a column vector. In your example of $$ax^2+bx+c \mapsto 2ax+b$$ the column vector $$\begin{bmatrix}a\\b\\c\end{bmatrix}$$ is multiplied on the left by the matrix$$\begin{bmatrix}0&0&0\\2&0&0\\0&1&0\end{bmatrix}$$