Embed Size (px)
Citation preview
Sections 1.8/1.9: Linear Transformations
€
x1a1 + x2a2 +L + x pa p = b€
Ax = b
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
€
x
€
Ax
€
Ax = b
Example:
1120
0032
0
0
1
2
€
= 2-2
0
⎡
⎣ ⎢
⎤
⎦ ⎥+1
3
2
⎡
⎣ ⎢
⎤
⎦ ⎥+ 0
0
−1
⎡
⎣ ⎢
⎤
⎦ ⎥+ 0
0
1
⎡
⎣ ⎢
⎤
⎦ ⎥=
−1
2
⎡
⎣ ⎢
⎤
⎦ ⎥
42
€
R4
A
€
x =
€
b
€
R2
1120
0032
1
2
0
1
2
1120
0032
Undefined
Undefined
€
R2
42
42
Recall that Ax is only defined if the number of columns of A equals the number of elements in the vector x.
€
R3
0
0
2
1
1120
0032
42
2
1
2
1
0
0
2
1
1120
0032
So multiplication by A transforms into .
A
€
x
€
R4
€
R2
€
R4
€
R2
€
x
€
b
€
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
Domain Codomain
€
Rn
€
Rm
€
T : Rn →Rm
T
Range
ADomain Codomain
€
Rn
Matrix Transformation:
x b
Let A be an mxn matrix.
€
a11 a12 ... a1n
a21 a22 ... a2n
... ... ... ...
am1 am2 ... amn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
⋅
x1
x2
...
xn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
=
a11x1 + a12x2 + ...+ a1n xn
a21x1 + a22x2 + ...+ a2n xn
...
am1x1 + am2x2 + ...+ amn xn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
A
€
x
€
Ax = b
€
=b€
Rm
Example: The transformation T is defined by T(x)=Ax where
For each of the following determine m and n.
71
53
31
1. A
010
0012. A
10
313. A
€
T : Rn →Rm
x
A x = b
A
Domain Codomain
€
Rn
€
Rm
Matrix Transformation:
nm b
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.
Linear Transformation:
Theorem: If T is a linear transformation, thenT(0)=0 andT(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
such that and . With no additional
information, find a formula for the image of an arbitrary x in R2.
1
2
0
1T
1
0
1
0T
€
x =x1
x2
⎡
⎣ ⎢
⎤
⎦ ⎥
1
0
0
121 xx
€
⇒ T x( ) =
1
0
0
121 xxT
1
0
0
121 TxTx
1
0
1
221 xx
2
1
11
02
x
x
2
1
x
xT
1
2
0
1T
1
0
1
0T
2
1
11
02
x
x
2
1
x
xT
1
4
1
2
11
02
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.
€
T : Rn →Rm
€
T x( ) = Ax
)( jeTnm
2
1
11
02
x
x
2
1
x
xT
je
1
2
0
1T
1
0
1
0T
2
1
x
x
2
1
x
xT
5
1
3
0
1T
5
1
2
1
0T
A is the standard matrix for the linear transformation T
5
1
2
5
1
3
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
10
01
10
01
01
10
01
10
10
01
Find the standard matrix of each of the following transformations.
HorizontalContraction &Expansion
VerticalContraction &Expansion
Projection ontothe x-axis
Projection ontothe y-axis
10
0k
k0
01
00
01
10
00
k
k
Applets for transformations in R2
From Marc Renault’s collection…Transformation of Pointshttp://webspace.ship.edu/msrenault/ggb/
linear_transformations_points.html
Visualizing Linear Transformationshttp://webspace.ship.edu/msrenault/ggb/
visualizing_linear_transformations.html
DefinitionA mapping is said to be onto if each b in is the image of at least one x in .
mn RRT :mR
mRnR
DefinitionA mapping is said to be one-to-one if each b in is the image of at most one x in .
mn RRT :mR
nR
Theorem 11Let be a linear transformation. Then, T is one-to-one iff has only the trivial solution. .
mn RRT :0)( xT
Theorem 12Let 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
mn RRT :
mR
