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.