Sections 1.8 and 1.9
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Transcript Sections 1.8 and 1.9
Sections 1.8/1.9:
Linear Transformations
Recall that the difference between the matrix equation
Ax b
and the associated vector equation
x1a1 x 2a 2 L x p a p b is just a matter of notation.
However the matrix equation Ax b 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 x by multiplication to produce a new vector
Ax
Example:
x =
A
2 3 0 0
0 2 1 1
2 4
b
2
-2 3 0 0 1
1
1 0 0
2
0 2 1
1 2
0
0
R4
R2
Recall that Ax is only defined if the number of columns
of A equals the number of elements in the vector x.
2 4
R2
2 3 0 0 2
0 2 1 1 1 Undefined
3
R
2 4
2
2 3 0 0 Undefined
0 2 1 1 1
0
2 4
R4
R2
1
2 3 0 0 2 1
0 2 1 1
2
0
0
A
b
x
R
R2
4
So multiplication by A transforms
x into b .
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.
Transformation:
Any function or mapping
T : R R
n
m
T
Range
Domain
R
Codomain
n
R
m
Matrix Transformation:
Let A be an mxn matrix.
Ax b
Domain
R
n
x
a11 a12 ...
a21 a22 ...
... ......
am1 am2 ...
A
b
Codomain
Rm
x b
A
a1n x1 a11 x1 a12 x 2 ... a1n x n
a2n x 2 a21 x1 a22 x 2 ... a2n x n
... ...
...
amn x n am1 x1 am 2 x 2 ... amn x n
Example: The transformation T is defined by T(x)=Ax where
T : R R
n
m
For each of the following determine m and n.
1 3
1. A 3 5
1 7
1 0 0
2. A
0
1
0
1 3
3. A
0
1
Matrix Transformation:
Ax=b
A
x
b
m n
Domain
R
n
Codomain
R
m
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.
Example. Suppose T is a linear transformation from R2 to R2
1 2
0 0
such that T and T . With no additional
0 1
1 1
information, find a formula for the image of an arbitrary x in R2.
x1
1
0
x x1 x2
x 2
0
1
1
0
T x T x1 x2
1
0
1
0
x1T x2T
0
1
2
0
x1 x2
1
1
x1 2 0 x1
T
x2 1 1 x2
1 2
T
0 1
0 0
T
1 1
x1 2 0 x1
T
x2 1 1 x2
4 2 0 2
1 1 1 1
Theorem 10.
Let T : R n R m be a linear transformation. Then there exists a
unique matrix A such that T x Ax for all x in Rn.
In fact, A is the m n matrix whose jth column is the vector T (e j )
where e j is the jth column of the identity matrix in Rn.
1 2
T
0 1
0 0
T
1 1
3
1
T 1
0 5
2
0
T 1
1 5
x1 2 0 x1
T
x2 1 1 x2
x1
T
x2
3
1
5
2
x1
1
x2
5
A is the standard matrix for the linear transformation T
Find the standard matrix of each of the following transformations.
Reflection through
the x-axis
1
0
Reflection through
the y-axis
1 0
0 1
Reflection through
the y=x
0
1
0
1
1
0
Reflection through
the y=-x
0 1
1
0
Reflection through
the origin
0
1
0 1
Find the standard matrix of each of the following transformations.
Horizontal
Contraction &
Expansion
k
k
Vertical
Contraction &
Expansion
k
0
0
1
1
0
0
k
Projection onto
the x-axis
1
0
0
0
Projection onto
the y-axis
0
0
0
1
Applets for transformations in R2
From Marc Renault’s collection…
Transformation of Points
http://webspace.ship.edu/msrenault/ggb/linear_transformations_points.html
Visualizing Linear Transformations
http://webspace.ship.edu/msrenault/ggb/visualizing_linear_transformations.html
Definition
m
A mapping T : R n R m is said to be onto R
n
if each b in R m is the image of at least one x in R .
Definition
A mapping T : R n R m is said to be one-to-one
n
m
R
R
if each b in
is the image of at most one x in
.
Theorem 11
Let T : R n R m be a linear transformation. Then,
T is one-to-one iff T ( x) 0 has only the trivial solution.
.
Theorem 12
Let T : R n R m be a linear transformation with standard
matrix A.
m
1. T is onto iff the columns of A span R .
2. T is one-to-one iff the columns of A are linearly independent