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.