ei is defined as the column vector where the ith element is 1 and all others are 0.
Ex. e2 in R3 is ⎝⎛010⎠⎞.
xT(x)=⎝⎜⎜⎛x1x2...xn⎠⎟⎟⎞=⎝⎜⎜⎛x10...0⎠⎟⎟⎞+⎝⎜⎜⎛0x2...0⎠⎟⎟⎞+...+⎝⎜⎜⎛00...xn⎠⎟⎟⎞=x1e1+x2e2+...+xnen=T(x1e1+x2e2+...+xnen)=T(x1e1)+T(x2e2)+...+T(xnen)=x1T(e1)+x2T(e2)+...+xnT(en)=[T(e1)T(e2)...T(en)]⎝⎜⎜⎛x1x2...xn⎠⎟⎟⎞=Ax.
The matrix A in this case is called the standard matrix for T.
This theorem tells us that all we don’t have to know what happens to every possible input vector to describe the linear transformation; we just have to know what happens to the ei vectors.
Ex. For some linear transformation T, the following are true:
T⎝⎛100⎠⎞T⎝⎛010⎠⎞T⎝⎛001⎠⎞=⎝⎛001⎠⎞=⎝⎛100⎠⎞=⎝⎛010⎠⎞
The standard matrix A of this transformation T is [T(e1)T(e2)T(e3)], which is just
⎝⎛001100010⎠⎞
based on the assertions above.
A linear transformation T:Rn→Rm is onto if, for any b in Rm, there is an x in Rn such that T(x)=b.
Ex. T:R3→R2, T(x)=(100100)x. Is T onto?
Yes, because this is equivalent to asking whether Ax=b will always have a solution. A has a pivot position in every row, so its columns span R2, so its associated function T is onto.
A linear transformation T:Rn→Rm is one-to-one if T(v1)=T(v2) implies that v1=v2 (i.e., its inverse will also be a function).
Using the linear transformation from the previous example, T⎝⎛123⎠⎞=(12) and T⎝⎛124⎠⎞=(12), but ⎝⎛123⎠⎞=⎝⎛124⎠⎞; therefore, T is not one-to-one.
If T(x)=0 has only the trivial solution, then T is one-to-one. This is the same definition as linear independence for the column vectors of the standard matrix of T!