Finding a basis

Dec 2019
1
0
Netherlands
Hello,

I have a linear algebra final coming up and I have a question about finding a basis. The vector space is defined as:

P = (a - c)x^3 + (b-c)x^2 + (a-b)x^ + b - c. A,b,c are element of C (complex numbers).

I really don't know how I can find a basis for this system.

I would appreciate some help.
 

SDK

Sep 2016
704
469
USA
Here is a hint: Let $T: \mathbb{C}^3 \to \mathbb{C}^4$ be the function which sends a triple, $(a, b, c)$ to the vector $(A, B, C, D)$ where $P$ is the polynomial $A + Bx + Cx^2 + Dx^3$. Its not hard to see that $T$ is a linear transformation between vector spaces. Can you represent $T$ as a matrix with respect to the standard basis? The rank of this matrix is the dimension of the vector space you want and a basis for the columns of the matrix will be a basis for this space.