Page 309 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(3)
for all vectors x in . This will show that T is multiplication by A and therefore linear. But before we
can produce this matrix, we need to observe that property (a) can be extended to three or more terms;
for example, if u, v, and w are any vectors in , then by first grouping v and w and applying property
(a), we obtain

More generally, for any vectors , , …, in , we have

Now, to find the matrix A, let , , …, be the vectors

and let A be the matrix whose successive column vectors are , , …, ; that is, (4)
If (5)

is any vector in , then as discussed in Section 1.3, the product is a linear combination of the
column vectors of A with coefficients from x, so

which completes the proof.

Expression 5 is important in its own right, since it provides an explicit formula for the standard matrix of a linear operator
in terms of the images of the vectors , , …, under T. For reasons that will be discussed later, the vectors ,

, …, in 4 are called the standard basis vectors for . In and these are the vectors of length 1 along the coordinate axes
(Figure 4.3.4).