# Linear Transformation: B-matrix [T]B

1. Dec 6, 2011

### PirateFan308

1. The problem statement, all variables and given/known data
Let V be polynomials, with real coefficients, of degree at most 2. Suppose that $T:V→V$ is differentiation. Find the $B$-matrix [T]B if B is the basis of V
B = {1+x, x+x2, x}

2. Relevant equations
For $T:V→V$ the domain and range are the same

[T]B is the matrix whose i-th column is $[T(vi)]_B$

$[T(v)]_C = A[v]_B$ where $A=[T]_B$

3. The attempt at a solution
So because the degree can be at most 2, the polynomials will be of the form a+bx+cx2. This can be denoted using a(1+x)+c(x+x2)+(b-a-c)(x). It will turn into a+bx (because we take the derivative, we take powers to a max of 1) and we would say a(1+x)+0(x+x2+b(x). After this, I'm not sure how to find the B-matrix, as I'm a bit confused as to what it is exactly.

2. Dec 6, 2011

### spamiam

Well, you wrote down the formula. Here $v_i$ is a basis vector. So take T of each of your basis vectors, and then express $T(v_i)$ as a linear combination of the basis vectors.

3. Dec 7, 2011

### HallsofIvy

Staff Emeritus
What is T(1+ x)? What is T(x+ x^2)? What is T(x)?
Write each of those as a linear combination of 1+ x, x+ x^2, and x and the coefficients are the columns of your matrix.