1. Sections 1.8/1.9: Linear Transformations

2. Recall that the difference between the matrix equation
and the associated vector equation
is just a matter of 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 Ax is only defined if the number of columns of A equals the number of elements in the vector x.

5. A
So multiplication by A transforms into

6. In the previous example, solving the equation Ax = b can be thought of as finding all vectors x in R4 that are transformed into the vector b in R2 under the “action” of multiplication by A.

7. Transformation:
Any function or mapping T
Range
Domain
Codomain

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

12. Example. Suppose T is a linear transformation from R2 to R2
such that and With no additional
information, find a formula for the image of an arbitrary x in R2.

14. Theorem 10.
Let be a linear transformation. Then there exists a unique matrix A such that for all x in Rn.
In fact, A is the matrix whose jth column is the vector where is the jth column of the identity matrix in Rn.
A is the standard matrix for the linear transformation T

15. Find the standard matrix of each of the following transformations.
Reflection through
the x-axis
Reflection through
the y-axis
Reflection through
the y=x
Reflection through
the y=-x
Reflection through
the origin

16. Find the standard matrix of each of the following transformations.
Horizontal
Contraction &
Expansion
Vertical
Contraction &
Expansion
Projection onto
the x-axis
Projection onto
the y-axis

17. Applets for transformations in R2
From Marc Renault’s collection…
Transformation of Points
Visualizing Linear Transformations

18. Definition
A mapping is said to be onto
if each b in is the image of at least one x in
Definition
A mapping is said to be one-to-one
if each b in is the image of at most one x in
Theorem 11
Let be a linear transformation. Then,
T is one-to-one iff has only the trivial solution
Theorem 12
Let be a linear transformation with standard matrix A.
1. T is onto iff the columns of A span
2. T is one-to-one iff the columns of A are linearly independent