386 CHAPTER 18. LINEAR FUNCTIONS

Example 18.2.15 Find the solution to

 1 2 10 2 11 1 1

 xyz

=

 211

 .

Add in the last column as above. 1 2 1 20 2 1 11 1 1 1

Now do row operations till you can see the answer. 1 0 0 1

0 1 0 10 0 1 −1

Thus x = 1,y = 1,z =−1.

Example 18.2.16 Find the solution to

 1 1 1−1 0 −31 2 −1

 xyz

=

 103

 .

Add in the last column  1 1 1 1−1 0 −3 01 2 −1 3

Now do row operations till you see the answer. Knowing when to stop is discussed more alittle later.  1 0 3 0

0 1 −2 00 0 0 1

The original system of equations has the same solution as one which includes the equation0x+ 0y+ 0z = 1 so there is no solution to this system of equations. The equations areinconsistent.

Definition 18.2.17 Let A =(a1 · · · an

). Then ak is linearly related to the

other columns means there are numbers xi such that ak = ∑i̸=k xiai.

This is just a more general notion than finding the solution to a system of equations inwhich you obtain a linear combination of columns of A equal to b in Ax= b. All that ishappening here is to note that there is nothing sacred about the last column in (A|b). Youcan ask the same question about all the other columns, whether they are a linear combina-tion of the other columns. It turns out that row operations preserve all linear relations.

Lemma 18.2.18 Let A and B be two m×n matrices and suppose B results from a rowoperation applied to A. Then the kth column of B is a linear combination of the i1, · · · , ircolumns of B if and only if the kth column of A is a linear combination of the i1, · · · , ircolumns of A. Furthermore, the scalars in the linear combination are the same. (The linearrelationship between the kth column of A and the i1, · · · , ir columns of A is the same as thelinear relationship between the kth column of B and the i1, · · · , ir columns of B.)