Week #4: Section 1.8, 1.9
Section 1.8: Linear Transformations
De…nition 1 A linear transformation is a mapping (or function) T from Rn to Rm satisfying (i) T (~u + ~v ) = T (~u) + T (~v ) and (ii) T ( ~u) = T (~u) for any real number :
Example 2 In 1-D, T (x) = cx (c is a constant) is a linear transformation. But T (x) =
ax + b is NOT (called a¢ ne transformation).
Any m n matrix Am
any ~u 2 Rn
n
de…nes a linear transformation T from Rn to Rm as follows: for
T (~u) = (Am
(1)
un 1 ) :
n ) (~
n
m
We can show that for any linear transformation T from R to R ; there is a m n matrix
Am n such that (1) holds. In other words, any linear transformation can be de…ned by a
matrix.
r 0
Example 3 (a) A =
is called a dilation if r > 1 and is contraction if 0 < r < 1: For
0 r
x
any ~u =
;
y
r 0 x
x
A~u =
=r
= r~u:
0 r y
y
(b) A =
0
1
For instance, A
1
is called a rotation ( rotation counter-clockwisely by ):
0
2
1
=
2
A~u =
0
1
(A~u) ~u =
x
y
1
0
x
=
y
y
= 0:
x
2
1
y
2
1
0
1
-1
-2
1
2
x
y
;
x
(c) A =
1 1
is a shear:
0 1
A~u =
1 1
0 1
x
x+y
x
y
y
=
=
+
= ~u +
:
y
y
y
0
0
For instance,
1
1
2
2
2
2
=
+
; A
=
+
;
2
2
0
2
2
0
A
y
2.0
1.5
1.0
0.5
0.0
0
1
2
2
3
4
x
Section 1.9: The Matrix of a Linear Transformation on Rn
For any m
n matrix A; we can de…ne a linear transformation T as
T : Rn ! Rm ; T (x) = Ax:
In this section, we shall see that the reverse is also true, i.e., any linear transformation T
can be represented by a A in the way de…ned above.
Example 1. Let
1
0
e1 =
; e2 =
0
1
and de…ne as
2
3
2 3
5
3
4
5
4
7 ; T (e2 ) = 8 5
T (e1 ) =
2
0
T
So T (x) = Ax; where
x1
x2
= T (x1 e1 + x2 e2 ) = x1 T (e1 ) + x2 2T (e2 )
2 3
2 3
5
3
4
5
4
7 + x2 8 5
= x1
2
0
3 2
3
2
5
3
5x1 3x2
x
= 4 7x1 + 8x2 5 = 4 7 8 5 1 = Ax:
x2
2
0
2x1
2
5
4
7
A=
2
3
3
8 5:
0
Theorem: Let T : Rn ! Rm be a linear transformation. Then there exists a m
matrix A such that
T (x) = Ax:
Furthermore, the matrix A has the form
A = [A (e1 ) A (e2 ) ::: A (en )]
and is called the standard matrix of T:
Example 2. Find the standard matrix for T (x) =
3
x2
:
x1
n
Sol:
T (e1 ) =
0
; T (e2 ) =
1
1
0
; A = [T (e1 ) T (e2 )]
0
1
1
:
0
What is the geometric explanation of this T ? (rotation by 90 degree counterclockwise).
De…nition: T : Rn ! Rm is be a linear transformation.
T is called one-to-one if T (x) = T (y) i¤ x = y: In other words, for any b; there is
at most one solution for T (x) = b. T is one-to-one i¤ T (x) = 0 has only the trivial
solution x = 0:
T is called onto Rm i¤ for any b in Rm ; there exists at least one solution f or T (x) = b:
In other words, T is onto Rm if the image of T …ll in all Rm :
In terms of matrices, we have
Theorem: Let T be a linear transformation with the standard matrix A:T hen
T is one-to-one i¤ the columns of A are linearly independent.
T is onto Rm i¤ the columns of A span Rm
Example 3. Let T (x1 ; x2 ) = (3x1 + x2 ; 5x1 + 7x2 ; x1 + 3x2 ) : Determine if T is one-toone and/or onto R3 :
Sol: Since
T (1; 0) = (3; 5; 1) ; T (0; 1) = (1; 7; 3) ;
we have its standard matrix
Its columns are
2
3
3 1
A = 45 75 :
1 3
2 3 2 3
3
1
455 ; 475 :
1
3
They are apparently linearly independent. So T is one-to-one. Next, A has at most two
pivot positions, T is not onto Rm :
Geometric interpretations
Rotation by angle ' counterclockwise:
A=
cos '
sin '
4
sin '
cos '
Re‡ection about x1
axis
A=
Re‡ection about x2
1
0
0
1
axis
A=
1 0
0 1
Re‡ection about the bisectional line x1 = x2
A=
Re‡ection about the line x1 =
0 1
1 0
x2
A=
0
1
1
0
A=
1
0
0
1
Re‡ection about the origin
Horizontal contraction/expansion ( k < 1 contraction, k > 0 expansion)
A=
k 0
0 1
Vertical contraction/expansion ( k < 1 contraction, k > 0 expansion)
A=
1 0
0 k
Horizontal shear (k > 0 to right, k < 0 to left)
A=
1 k
0 1
Vertical shear (k > 0 to up, k < 0 to down)
Projection onto x1
A=
1 0
k 1
A=
1 0
0 0
axis
5
Projection onto x2
axis
A=
Homework:
Section 1.8: 1, 3, 9, 11, 15, 17
Section 1.9: 3, 7, 9, 17, 19, 25, 27
6
0 0
0 1