9.2. CONSTRUCTING THE MATRIX OF A LINEAR TRANSFORMATION 187

Definition 9.1.3 A linear transformation is called one to one (often written as 1−1) if itnever takes two different vectors to the same vector. Thus T is one to one if whenever x ̸= y

T x ̸= T y.

Equivalently, if T (x) = T (y) , then x = y.

In the case that a linear transformation comes from matrix multiplication, it is com-mon usage to refer to the matrix as a one to one matrix when the linear transformation itdetermines is one to one.

Definition 9.1.4 A linear transformation mapping Fn to Fm is called onto if whenevery ∈ Fm there exists x ∈ Fn such that T (x) = y.

Thus T is onto if everything in Fm gets hit. In the case that a linear transformationcomes from matrix multiplication, it is common to refer to the matrix as onto when thelinear transformation it determines is onto. Also it is common usage to write TFn, T (Fn) ,orIm(T ) as the set of vectors of Fm which are of the form T x for some x ∈ Fn. In the casethat T is obtained from multiplication by an m×n matrix A, it is standard to simply writeA(Fn), AFn, or Im(A) to denote those vectors in Fm which are obtained in the form Ax forsome x ∈ Fn.

9.2 Constructing The Matrix Of A Linear Transforma-tion

It turns out that if T is any linear transformation which maps Fn to Fm, there is always anm×n matrix A with the property that

Ax = T x (9.1)

for all x ∈ Fn. Here is why. Suppose T : Fn 7→ Fm is a linear transformation and you wantto find the matrix defined by this linear transformation as described in 9.1. 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,

T x =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) .We state this as the following theorem.