summary of ch6

8/11/2019 Summary of CH6

1/14

Every linear Transformation T maps n-dim

vectors to m-dim vectors

can be represented by a Matr ix AFurthermore

Tu=Au=multiplication of matrices A and u

Linear Transformation and matrix

Summary of lecture on 17 March, Linear transformation

1


2/14

Example

Linear transformation

114

11

4

T

1

4Ti i j 1

4Tj i j

T: 2-D vectors

2-D vectors

Then

If

2

1

4Ti i j

1

4Tj i j


3/14

Example

T: 3-D vectors 2-D vectors

2Ti i Tj i j Tk i j

2 1 1

0 1 1T

Linear

transformation

Then

If

3

2Ti i Tj i j

Tk i j


4/14

Volume and Determinant

Area of parallelogram

induced by u and v

v

u

=

=

A=

Let

Let which is the matrix

induced by vectors

u and v

= detA

det 0 iff vectors and vare on the same st line

A u u

v

iff iffa c

u vb d

4


5/14

Vol of parallelepiped

induced by u,v and w

Let A=

11 12 13

21 22 23

31 32 33

=

11

21

31

=

12

22

32

=

13

23

33

= detA

1 1

det 0 iff three vectors

, , are on the same st line ,

or are on the same plane,

A

u v w u v w

w u v

uv

w

u

u

w5


6/14

A transformation T from 3-D to 3-D has ONLY three

cases:

Case1: det(T) not zero

T mapsthe whole 3-d space ONTO the whole 3-d space

Image of T from 3D to 3D

TGiven any point

or vector in 3D space

d

e

f

we can find such thatab

c

a d

T b e

c f

6


7/14

Case 2: det(T)=0

T maps the whole 3-d space

ONTO a plane (the whole plane)passing through origin

Subcase 1

T

7


8/14

Subcase 2 T maps the whole 3-d spaceONTO a line (the whole line)

passing through origin

T

8


9/14

9

AX=B and linear mapping

Consider linear system of three equations

AX Bor more precisely 11 12 13

21 22 23

31 32 33

a a a

a a aa a a

1

2

3

x

xx

1

2

3

b

bb

Now we may consider: matrix A maps a point

1

2

3

x

x

x

to a point1

2

3

b

b

b


10/14

10

Suppose det 0A Then we know that for any

1

2

3

b

b

b

We can find

1

2

3

x

x

x

such that

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

1

2

3

x

x

x

1

2

3

b

b

b

Hence matrix A maps the whole 3D space ONTO

the whole 3D space


11/14

Suppose Det 0A

Then the matrix A maps the whole 3D space

onto a plane or onto a line

A(whole 3D space) is call

the image of 3D space under mapping A

Given a point

1

2

3

bb

b

If this point is notin the image

Then we cant find1

2

3

xx

x

such that A1

2

3

x

x

x

1

2

3

b

b

b

It means this system of equations has no solution for this case


12/14

12

Suppose11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

A

Then

11

21

31

a

Ai aa

12

22

32

a

Aj aa

13

23

33

a

Ak aa

Suppose Det 0A The image of the whole 3D space depends

on the relation between Ai Aj Ak


13/14

13

If Ai Aj Ak

then the image is a line passing through origini.e., A maps the whole 3D space onto a line

1 1 1

1 1 1

0 0 0

A 11

0

Ai Aj Ak

Example

A maps the whole 3D space

onto a line passing through (0,0,0) and (1,1,0)


14/14

14

If 1 1Ak Ai Aj then the image is a plane passing through origin

i.e., A maps the whole 3D space onto a plane

2 1 11 2 1

0 0 0

A 1 11 2 1 2 11 11 1 2 1 2

3 30 0 0 0 0

Example

i.e., A maps the whole 3D space onto a plane

induced by

1 1

2 1

1 2

0 0

summary of ch6

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