Here's a written summary of today's lecture. I wrote this up because in Points 1 and 5 below what I said in class went beyond the textbook. Given: spaces U, W and a basis u_1,...,u_m of U. 0. L(U,W) is the set of all linear transformations from U to W. Transformation = function; linear = additive + homogeneous. 1. In the case U=W=R^2 (or R^3), we can understand elements of L(U,W) geometrically. 2. A linear transformation T from U to W is completely determined by T(u_1),...,T(u_m). 3. Defining a specific linear transformation T from U to W is easier than it looks. Simply choose T(u_1) to be some v_1 in W, T(u_2) to be some v_2 in W, and so on. There is one and only one linear transformation T satisfying T(u_i)=v_i for all i=1,...,m: T(a_1*u_1+...+a_m*u_m)=a_1*v_1+...a_m*v_m for all a_1,...,a_m in F. 4. L(U,W) becomes a vector space if we define functions S+T and c*S by the rule (S+T)(u)=S(u)+T(u) and (c*S)(u)=c*S(u). 5. If W also has a basis w_1,...,w_n, then L(U,W) has a finite basis, namely, the list of all T in L(U,W) with the property that, for some i, T(u_i) equals some w_j, but T(u_i)=0 for all other i. (These basis elements correspond to n by m matrices that are full of zeroes except for one one. This correspondence will be made precise later in chapter 3.) 6. Because of Point 2, if S and T are in L(U,W), then proving S=T is easier than it looks. ("S=T" means S(u)=T(u) for all u in U.) We just need to prove S(u_i)=T(u_i) for each i.