1. Section 1.8:
Introduction to Linear Transformations

2. Recall that the difference between the matrix equation
and the associated vector equation
is notation.
However, the matrix equation can arise is linear algebra (and applications) in a way that is not directly connected with linear combinations of vectors.
This happens when we think of a matrix A as an object that acts on a vector by multiplication to produce a new vector

3. Example: A =

4. Recall that is only defined if the number of columns of A equals the number of elements in .

5. A
So multiplication by A transforms into

6. In the previous example, solving the equation can be thought of as finding all vectors in that are transformed into the vector in under the “action” of multiplication by A.

7. Transformation:
Function or Mapping T T
Range
Domain
Codomain

8. Let A be an mxn matrix.
Matrix Transformation:
Codomain A
Domain b x A

9. Example: The transformation T is defined by T(x)=Ax where
For each of the following determine m and n.

10. Matrix Transformation:
Ax=b x A b
Domain
Codomain

11. Linear Transformation:
Definition:
A transformation T is linear if
(i) T(u+v)=T(u)+T(v) for all u, v in the domain of T:
(ii) T(cu)=cT(u) for all u and all scalars c.
Theorem:
If T is a linear transformation, then
T(0)=0 and
T(cu+dv)=cT(u)+dT(v) for all u, v and all scalars c, d.