374 CHAPTER 18. LINEAR FUNCTIONS

Proposition 18.1.3 Let T : Rn → Rm be linear, T ∈ L (Rn,Rm). Then for

x=(

x1 · · · xn)T

,

T (x) = x1T (e1)+ x2T (e2)+ · · ·+ xnT (en)

In other words, for each i ≤ m,

T (x)i = x1T (e1)i + x2T (e2)i + · · ·+ xnT (en)i ≡ ∑k

xkT (ek)i

Proof: Since T is linear, T (x) = T (∑nk=1 xkek) = ∑

nk=1 xkT (ek) which is the above.

Note that Proposition 18.1.3 shows that if you know what the linear function does toeach ek, then you know what it does to an arbitrary vector x.

Example 18.1.4 Suppose T : R3 → R3 is linear and

Te1 ≡

 111

 ,Te2 ≡

 123

 ,Te3 ≡

 101

Describe T (x) .

According to the above proposition,

Tx= x1

 111

+ x2

 123

+ x3

 101

There is a shortened version of this described in the following definition.

Definition 18.1.5 Letting x be a vector in Rn, and letting u1, · · · ,un be vectors inRm, the linear combination

x1u1 + x2u2 + · · ·+ xnun ≡n

∑k=1

xkuk

is written as

(u1 u2 · · · un

)

x1x2...

xn

Here

(u1 u2 · · · un

)is called an m× n matrix, meaning it is a rectangular array

of numbers having m rows (rows are horizontal) and n columns (columns are vertical). Thekth column from the left will be uk. Note that a linear combination is just an expressionconsisting of scalars times vectors added together. For T a linear transformation, its matrixA is such that T (x) = Ax.

Theorem 18.1.6 Let T ∈ L (Rn,Rm) . Then the matrix of T is the following whereeach Tek is a column vector: (

Te1 Te2 · · · Ten)