Systems of linear equations , with matrix form
Ax = b, are often usefully analyzed by viewing the
equation as the problem that asks for an unknown
input x for a function that produces a known output
b. The rule for this function is the one that takes
vector inputs x ∈ Rn and returns vector outputs
Ax ∈ Rm. We call such functions with vector
inputs and vector outputs transformations (of
Euclidean spaces).
Using standard notation for describing functions,
we can refer to such a transformation as a function,
or map, of the form T:Rn →Rm. Here, T is the
name of the transformation that carries vector
inputs from Rn to vector outputs from Rm according
to some well-defined rule. In the context of systems
of linear equations (the one of prime interest to us),
we could define T by the formula T (x) = Ax where
A is some m×n matrix. See Example 1, p. 74, for
details, and study the marginal diagrams there for
a glimpse at how one might form a mental picture
of such a transformation.
Examples 2, 3, 4, and 5 on pp. 76-78 illustrate that
much geometrical information is captured by the
behavior of certain transformations of the form
T(x) = Ax. It is not coincidence that these
transformations obey the simple properties
T(u+ v) = T(u) +T(v) for all inputs u, v;
T(cu) = cT(u) for all inputs u and scalars c.
Any transformation of Euclidean spaces that
satisfies these two properties is called a linear
transformation. The defining properties can be
inter preted as saying that linear transformations
preserve vector additions and scalar multiplications .
Because of the algebraic properties of the matrix-vector
product , it is clear that all transformations
of the form T(x) = Ax are automatically linear
transformations. But as we shall soon see, the
converse statement is also true: every linear
transformation has a matrix form T(x) = Ax!
Other important properties of linear
transformations:
Theorem Any linear transformation T:Rn →Rm
takes the zero vector in Rn to the zero vector in Rm.
Proof T(0) = T(0 + 0) = T(0) +T(0) =>T(0) = 0. //
Theorem The transformation
T:Rn →Rm is linear
if and only if for all vectors
u,v ∈ Rn and all scalars
c, d, the relation
T(cu+dv) = cT(u)+ dT(v) holds.
Proof If T is a linear transformation, then, using
the defining properties, we have
T(cu+dv) = T(cu) +T(dv) = cT(u) + dT(v).
Conversely, if T is a transformation for which
T(cu+dv) = cT(u)+ dT(v)
holds, then setting
c = d = 1 shows that
T(u+ v) = T(u) +T(v) for all u and v in
Rn; and
setting
d = 0 shows that
T(cu) = cT(u) for every
choice of c and u. So T must be linear. //
Repeated application of the property
T(cu+dv) = cT(u)+ dT(v), shows that if T is a
linear transformation, then for any collection of
vectors 
and associated scalars
That is, T carries any linear combination of a set
of vectors in Rn to the same linear
combination of their images 
in
Rm. This notion is often referred to as the
superposition principle.
We are now in a position to prove the theorem
alluded to earlier:
Theorem Any linear transformation T:Rn →Rm
has an associated m×n matrix A for which
T(x) = Ax. More specifically,
is the matrix whose jth column is the image 
of the vector 
that is the jth column of the
n × n
identity matrix
Proof Since 
is the
identity matrix,
So, by the linearity of T,
The matrix A obtained by this theorem is called the
standard matrix for the transformation T.
For instance, the tables on pp. 85-87 present the
standard matrices for the linear transformations
from R2 to R2 which represent reflections across
certain lines, reflection through a point (the
origin), certain dilations and shears, and
projections onto certain lines.
It is significant to note that the geometric
transformations in Tables 1-3 (pp. 85-86) are one-to-one
as functions, and they map R2 onto R2. In
contrast, the projection maps in Table 4 (p. 87) are
neither one-to-one nor onto. The properties of
being one-to-one and onto are related to ideas we
have explored earlier; this is spelled out in the
fol lowing two theorems :
Theorem The linear transformation T:Rn →Rm is
one-to-one if and only if the zero vector in Rn is the
only vector that is mapped by T to the zero vector
in Rm, i.e., T(x) = 0 has only the trivial solution.
Proof Since T is a linear transformation, T(0) = 0.
So, if T is one-to-one, T(x) = 0 can have only the
trivial solution x = 0.
Conversely, suppose T is a linear
transformation for which T(x) = 0 has only the
trivial solution. Then, if u and v are vectors in Rn
for which T(u) = T(v), it follows that
T(u− v) = T(u) −T(v) = 0
from which we conclude that u− v = 0, or u = v.
Therefore, T is one-to-one. //
Theorem Suppose the linear transformation
T:Rn →Rm has standard matrix A. Then
(1) T is one-to-one if and only if the columns of A
are linearly independent; and
(2) T is onto if and only if the columns of A span
Rm.
Proof (1) is an immediate consequence of the
previous theorem, since we know that the columns
of A are linearly independent if and only if Ax = 0
has only the trivial solution.
To prove (2), recall the theorem that says that
the columns of A span Rm if and only if the
equation Ax = b is consistent for every b ∈ Rn.
But this holds if and only if T(x) = Ax is an onto
function. //
