Quote:

Originally Posted by

**HallsofIvy** But to generalize: If L is a linear transformation from vector space U, with dimension n, to vector space V, with dimension m, then it can be written in matrix form for given basis $\displaystyle \{u_1, u_2, ..., u_n\}$ of U and given basis $\displaystyle \{v_1, v_2, ..., v_m\}$ of V (changing bases will result in different but "equivalent" matrices representing the same transformation).

Take $\displaystyle L(u_1)$ and write it as a linear combination of the basis $\displaystyle \{v_1, v_2, ...v_n\}$, which you can do since $\displaystyle L(u_1)$ is in V. Say, $\displaystyle L(u_1)= a_1v_1+ a_2v_2+ ...+ a_mv_m$. Then $\displaystyle \begin{bmatrix}a_1 \\ a_2 \\ \cdot \\\cdot \\\cdot \\ a_m \end{bmatrix}$ is the first column in the matrix. Similarly, writing $\displaystyle L{u_2}$ as a linear combination of $\displaystyle v_1, v_2, ..., v_m$ gives the second column, etc.

I'm new to a lot of this stuff so some of that definition didn't really make sense to me... You are saying that i would have to approach a problem like the one below differently, correct?

(b) Let

L : R3 ->R2 be the linear transformation defined by

L

(x, y, z) = (x + y + z, y + z)

(i) Find the standard (2 × 3) matrix representation of L and compute L(0,−1, 1).