134 CHAPTER 8. MATRICES

Example 8.2.4 Write the system of equations

x+2y− z = 2x−3y+ z = 1

in the form Ax= b.

According to the above, this system can be written as

(1 2 −11 −3 1

) xyz

=

(21

)

The following is the most fundamental observation about multiplying a matrix times avector.

Theorem 8.2.5 Let A be an m× n matrix and let x,y be two vectors in Fn with a,b twoscalars. Then

A(ax+by) = aAx+bAy

Proof: By the above definition and the way we add vectors,

(A(ax+by))i = ∑j

Ai j (ax j +by j) = a∑j

Ai jx j +b∑j

Ai jx j

= a(Ax)i +b(Ay)i = (aAx+bAy)i

Since the ith entries coincide, it follows that A(ax+by) = aAx+bAy as claimed. ■

Definition 8.2.6 Define some special vectors ei as follows:

ei ≡

1 in the ith position︷ ︸︸ ︷(0 · · · 0 1 0 · · · 0

)T

Thus in F3, we would have

e1 =

 100

 ,e2 =

 010

 ,e3 =

 001

Observation 8.2.7 Let A be an m×n matrix. Then for ei ∈ Fn,Aei delivers the ith columnof A. To see this,

(Aei)k = ∑j

Ak j (ei) j = Aki

because (ei) j = 0 unless j = i when it is 1. Thus, for k arbitrary, the kth entry of Aei is Aki.Thus the result of multiplying by ei is(

A1i A2i · · · Ani

)T