# Lisa's question at Yahoo! Answers (Matrix of a linear map)

#### Fernando Revilla

##### Well-known member
MHB Math Helper
Here is the question:

Let V be the space spanned by the two functions cos(t) and sin(t). Find the matrix A of the linear transformation T(f(t)) = f''(t)+3f'(t)+4f(t) from V into itself with respect to the basis {cos(t),sin(t)}.
Here is a link to the question:

Hello Lisa, we have: $$T(\cos t)=(\cos t)''+3(\cos t)'+4\cos t=-\cos t-3\sin t+4\cos t=3\cos t-3\sin t\\T(\sin t)=(\sin t)''+3(\sin t)'+4\sin t=-\sin t+3\cos t+4\sin t=3\cos t+3\sin t$$ Transposing coefficients: $$A=\begin{bmatrix}{3}&{3}\\{-3}&{3}\end{bmatrix}$$