Download §1.8 Introduction to Linear Transformations Let A = [a 1 a2 an] be

§1.8 Introduction to Linear Transformations
Let A = [a1 a2 . . . an ] be an m x n matrix whose columns are the vectors a1 , a2 , . . . , an in Rm . If x is a vector in Rn we
define the product Ax as a linear combination of the columns of A using the (scalar) entries in x as weights:
 
x1
 .. 
Ax = [a1 a2 · · · an ]  .  = x1 aa + x2 a2 + · · · + xn an = y
xn
Since the columns of A live in Rm so does y = x1 aa + x2 a2 + · · · + xn an . So we take a vector x in Rn and multiply it
on the left by a given m by n matrix A to produce a unique vector y in Rm . We have just created a function from Rn
to Rm !
dfn: A transformation or function or mapping T from Rn to Rm is a rule that assigns to each vector x in Rn a
unique (meaning one and only one) vector y in Rm . The set Rn is called the domain of T and the set Rm is called the
codomain of T . The notation is T : Rn → Rm .
For x in Rn we call T (x) in Rm the image of x under the action of T or just the image of x under T . Sometimes we
write x 7→ T (x). The set of all images T (x) in Rm is called the range of T .
Every transformation T in this section is enacted through matrix multiplication. For each x in Rn we compute T (x) as
Ax, where A is an m x n matrix. We sometimes write x 7→ Ax. The range of T consists of all linear combinations of
the columns of A, i.e. the range of T is the span of the columns of A.
Typical problems given an m x n matrix A and associated transformation T : Rn → Rm defined by T (x) = Ax.
• For a given vector u in Rn find T (u).
• Can you find a vector x in Rn whose image under T is a given vector b in Rm .
• Is there more than one vector x in Rn whose image under T is a given vector b in Rm ?
• Determine if a given vector b is in the range of T .
• Is the range of T all of Rm , i.e. do the columns of A span Rm ?
dfn: A transformation (or mapping) T is linear (and T is called a linear transformation) if for all vectors u and v in
the domain of T and all scalars c,
(i) T (u + v) = T (u) + T (v)
(ii) T (c u) = c T (u)
Recall Theorem 5:
(T preserves addition)
(T preserves scalar multiplication)
If A is an m x n matrix and u and v are vectors in Rn and c is a scalar, then:
(a) A(u + v) = Au + Av
(b) A(cu) = c(Au)
Hence if A is an m × n matrix, then by Theorem 5, T : Rn → Rm given by T (x) = Ax is always linear.
Facts: If T is a linear transformation:
1. T (0) = 0
(NB: This means that if T (0) 6= 0 then T is not linear.)
2. T (c u + d v) = c T (u) + d T (v) for all vectors u and v in the domain of T and all scalars c and d.
3. In general: T (c1 v1 + c2 v2 + · · · + cp vp ) = c1 T (v1 ) + c2 T (v2 ) + · · · + cp T (vp )