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Linear Algebra
Linear Transformations
2
Real Valued Functions
Function from to R
Function from to R
Function from to R
Function from R to R
Formula Example Description
f x
f x , y
f x , y , z
f x 1 , x 2 ,, xn
f x x^2
f x , y x^2 y^2
f x , y , z x^2 y^2 z^2
2
2 2
2 1 , 2 , , 1
n
n x
fx x x x x
R^2
R^3
R^ n
3
Linear Transformations
To illustarte one important way in which transformations can arise, suppose that are real-valued functions of n real variables, say
These m equations assign a unique point in to each point in.
f (^) 1 , f 2 ,, f m
m m ^ n
n
n
w f x x x
w f x x x
w f x x x
1 2
2 2 1 2
1 1 1 2
w 1 , w 2 ,, wm
R^ m ^ x 1^ ,^ x 2 ,^ , xn Rn
4
Introduction
Another way to view : Matrix A is an object acting on x by multiplication to produce a new vector Ax or b. Example
Ax = b
Ax = b
5
Introduction
Solving Ax = b amounts to finding all vectors x in R n that are transformed into the vector b in Rm^ under the action of multiplication by Am×n. The correspondence from x to b is a function from one set of vectors to another. This concept is generalization of f:R -> R.
x (^) b
Multiplication by A
A m×n x = b :
=> T n^ m
6
Introduction
A transformation (or function or mapping) T from R n^ to R m^ is a rule that assigns to each vector x in R n , a unique vector T(x) in R m. The vector T(x) is called the image of x under T. The set R n^ is called the Domain of T , and the set of all images T(x) in R m^ is called Range. We write T:Rn^ -> R m^ to indicate that the domain of T is R n^ and the range of T is contained in R m.
7
Introduction
8
Matrix Transformation
For each x in R n^ , Tx is computed as Ax , where A is an m×n matrix. We sometimes denote this Matrix Transformation by x -> Ax. Domain of T lies in R n when A has n columns and range of T lies in R m^ when each column of A has m entries.
13
c. Any x whose image under T is b must satisfy
We have seen in (b) that there is exactly one x whose image is b.
1 2
x
x
^
^
^
Matrix Transformation – Example 1
14
d. The vector c is in the range of T if c is the image of some x in R 2 , that is, if c=T(x) for some x. This is just another way of asking if the system Ax=c is consistent. To find the answer, row reduce the augmented matrix
The third equation 0=-35 shows that the system is inconsistent. So c is not in the range of T.
Matrix Transformation – Example 1
15
Matrix Transformation –
Example 2
If
Then x -> Ax projects points in R^3 onto R2.
A
^ ^
1 1 2 2 3
x x x x x
x (^3)
16
Matrix Transformation
Let
Then the transformation T:R 2 -> R 2 defined by Ax is called the Shear Transformation. For example the image of
1 3 0 1
A ^
is
17
Matrix Transformation
Shear Transformation
18
Linear Transformations
Theorem If A is an m×n matrix, u and v are vectors in R n^ and c is a scalar, then a) A(u+v) = Au + Av b) A(cu) = c(Au) Proof For simplicity take n=3, so A =[a 1 , a 2 , a 3 ]
1 1 1 2 3 2 2 3 3
u v A u v a a a u v u v
^
19
Linear Transformations
1 1 1 2 2 2 3 3 3
1 1 1 1 2 2 2 2 3 3 3 3
1 1 2 2 3 3 1 1 2 2 3 3
1 1
1 2 3 2 1 2 3 2
3 3
a ( u v ) a ( u v ) a ( u v )
a u a v a u a v a u a v
a u a u a u a v a v a v
u v
a a a u a a a v
u v
Au Av
Similarly (b) can also be proved 20
Linear Transformations
A transformation T is linear, if
1. T(u+v) = T(u) + T(v) **(for all u,v in the domain of T)
- T(cu) = cT(u)** (for all u and all scalars c) Property (1) says that the result T(u+v) i.e. first adding u+v and then applying T is same as applying T first to u and v and then adding T(u) and T(v).
25
Linear Transformations – Example 2
Solution
Note that T(u+v) is obviously equal to T(u)+T(v). Also, it appears that T rotates u , v and u+v counterclockwise about the origin through 90 degrees.
0 1 4 1 0 1 2 3 ( ) , ( ) 1 0 1 4 1 0 3 2
0 1 6 4 ( ) 1 0 4 6
T u T v
T u v
(^) (^) (^)
^ (^)
26
Linear Transformations – Example 2
A rotation transformation
27
Linear transformation
29
Linear Transformation
30
Linear transformation
31
Composition of Transformation
32
Composition of Transformation
37
Linear transformation (Reflections)
38
Orthogonal Projections
39
Orthogonal Projections
40
Orthogonal Projections
41
Contractions & Dilations
42
Rotations
43
Rotations
44
Rotations
49
Example (Contd.)
50
Example (Contd.)
51
Example
52
Example
53
Example (Contd.)
54
Example (Contd.)
55
The Matrix of a Linear
Transformation
Every Linear Transformation T:Rn^ -> Rm^ is a matrix transformation x -> Ax.
The key to finding A is to observe that T is completely determined by what it does to the columns of the n×n identity matrix In.
56
The Matrix of a Linear
Transformation – Example 1
The columns of are.
Suppose T is a linear transformation from R 2 into R 3 such that
With no additional information, find a formula for the image of an arbitrary x in R 2.
2
1 0 0 1
I ^ 1 2
(^1) and 0 0 1
e ^ e
2
5 3 ( 1) 7 and ( ) 8 2 0
T e T e
^ ^ ^ ^
61
Find the standard matrix A for the dilation transformation T(x) = 3x, for x in R^2.
Solution
Write
1 1 2 2
( ) 3 and ( ) 3
T e e T e e
A
The Matrix of a Linear
Transformation – Example 1
62
Let T:R 2 -> R 2 be the transformation that rotates each point in R 2 about the origin through an angel , with counterclockwise rotation for a positive angle. Find the standard matrix A of this transformation. Solution rotates to
and rotates into
1 0
cos sin
sin cos
cos sin
sin cos
A
The Matrix of a Linear
Transformation – Example 2
=>
63
The Matrix of a Linear
Transformation – Example 2
A rotation transformation
64
Onto mapping
A mapping T:R n^ -> R m^ is said to be onto R m^ if each b in Rm^ is the image of at least one x in Rn.
65
Onto Mapping
Equivalently, T is onto When the range of T is all the codomain , there exists at least one solution of T(x)=b. The mapping T is not onto when there is some b in for which T(x)=b has no solution.
R^ m
Rm
Rm
66
One-to-one mapping
A mapping T:R n^ -> R m^ is said to be one-to-one if each b in Rm^ is the image of at most one x in Rn.
67
One-to-one mapping - Example
Let T be the linear transformation whose standard matrix is
Solution A happens to be in echelon form, we can see at once that A has a pivot position in each row. For each , the equation Ax=b is consistent. In other words, the linear transformation T maps ( its domain) onto However, since the equation Ax=b has free variables, each b is the image of more than one x , i.e., T is not one-to-one****.
R^4
1 4 8 1 0 2 1 3 0 0 0 5
A
^
b R^3 3
R
68
One-to-one mapping
Theorem Let T:Rn^ -> R m^ be a linear transformation. Then T is one-to-one if and only if the equation T(x)=0 has only the trivial solution.
73
Onto and one-to-one mapping -
Example
Alternatively Let be any typical vector from
then must be consistent if T has to be onto.
3
2
1
b
b
b b
R^3
3
2
1
1 2
b
b
b
k k
74
Onto and one-to-one mapping -
Example
The augmented matrix of the above system is
In general is nonzero. Hence the system is inconsistent. Thus T is not onto.
3
2
1
b
b
b
1
2
3
b
b
b
1 3
2 3
3
b b
b b
b
1 2 3
2 3
3
b b b
b b
b
b 1 b 2 2 b 3
R 13
3 1
2 1 3
R R
R R
R 3 R 2
75
Onto and One-to-one Mapping
Example
The Identity Transformation Let defined by
(a) T maps onto since the columns of A span T is one-to-one since the columns of A are linearly independent.
T : R^3 R^3 T^ ( x ) x
3
2
1
3
2
1
(x) x
x
x
x
x
x
x
T A
R^3 R^3
R^3
