56 CHAPTER 2. LINEAR TRANSFORMATIONS

Let Aik = eTi Lek, to prove the existence part of the theorem.To verify uniqueness, suppose Bx = Ax = Lx for all x ∈ Fn. Then in particular, this is

true for x = ej and then multiply on the left by eTi to obtain

Bij = eTi Bej = eTi Aej = Aij

showing A = B. ■

Corollary 2.3.5 A linear transformation, L : Fn → Fm is completely determined by thevectors {Le1, · · · , Len} .

Proof: This follows immediately from the above theorem. The unique matrix determin-ing the linear transformation which is given in 2.22 depends only on these vectors. ■

For a different proof of this theorem and corollary, see the following section.This theorem shows that any linear transformation defined on Fn can always be consid-

ered as matrix multiplication. Therefore, the terms “linear transformation” and “matrix”are often used interchangeably. For example, to say that a matrix is one to one, means thelinear transformation determined by the matrix is one to one.

Example 2.3.6 Find the linear transformation, L : R2 → R2 which has the property that

Le1 =

(2

1

)and Le2 =

(1

3

). From the above theorem and corollary, this linear trans-

formation is that determined by matrix multiplication by the matrix(2 1

1 3

).

2.4 Geometrically Defined Linear Transformations

If T is any linear transformation which maps Fn to Fm, there is always an m × n matrixA ≡ [T ] with the property that

Ax = Tx (2.23)

for all x ∈ Fn. What is the form of A? Suppose T : Fn → Fm is a linear transformationand you want to find the matrix defined by this linear transformation as described in 2.23.Then if x ∈ Fn it follows

x =

n∑i=1

xiei

where ei is the vector which has zeros in every slot but the ith and a 1 in this slot. Thensince T is linear,

Tx =

n∑i=1

xiT (ei)

=

 | |T (e1) · · · T (en)

| |



x1...

xn

 ≡ A

x1...

xn

and so you see that the matrix desired is obtained from letting the ith column equal T (ei) .This proves the existence part of the following theorem.