Page 599 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

is a basis for V, the vector can be written in the form

Since ,…, lie in the kernel of T, we have , so

Thus S spans the range of T.

Finally, we show that S is a linearly independent set and consequently forms a basis for the range of T. Suppose that some linear
combination of the vectors in S is zero; that is,

(2)

We must show that . Since T is linear, 2 can be rewritten as

which says that is in the kernel of T. This vector can therefore be written as a linear combination of the
basis vectors , say

Thus,

Since is linearly independent, all of the k's are zero; in particular, , which completes the proof.

Exercise Set 8.2

Click here for Just Ask!

Let be the linear operator given by the formula
1.

Which of the following vectors are in ?

(a)

(b)

(c)

Let be the linear operator in Exercise 1. Which of the following vectors are in
2.

(a)

(b)