Sense making in linear algebra

Lee Peng YeeBangkok10-07-2008

Historical events

• Geometry went algebraic after Felix Klein• Algebra turned abstract• Linear algebra came from geometry

Contents

• Vector spaces and bases• Matrices• Eigenvalues and eigenvectors

Two questions

• Why linear algebra or motivation• Why eigenvalues and eigenvectors

Why linear algebra

• Linear systems• Geometric transformations• Markov chains• Lately linear codes

LINEAR CODES

• Message encode transmit received decode detect error correct error final message

• Concepts used: vector space/linear space, basis, matrices, matrix multiplication

An example

• {000 000, 001 110, 010 101, 011 011,100 011, 101 101, 110 110, 111 000}a linear code or a linear space (closed under linear combination with 1 + 1 = 0)

• Elements in the space are codewords

Basis

• { 100 011, 010 101, 001 110 } forms a basis for the space{000 000, 001 110, 010 101, 011 011,100 011, 101 101, 110 110, 111 000}

• Check 000 000 = 100 011 + 100 011, 011 011 = 010 101 + 001 110 etc

Generator matrix

011101110

100010001

G

Message

• [110] is a 3-bit message• We turn it into a codeword (encode)• Transmit the codeword• Then decode

Encoding

011011011101110

100010001

011

Message received

• The codeword [110 110] is transmitted • Suppose the received word is

[100 110] (with an error)• [100 110] is not a codeword• How do we decode, detect the error and

correct it?

Parity check matrix

100010001

011101110

H

Decoding

000

011011

100010001

011101110

101

011001

100010001

011101110

Error detecting

• If HxT = [0 0 0]T then x is a codeword • If HxT = [1 0 1]T then en error is detected

• If x is a codeword, r is received word, and e is error then HrT = HxT + HeT = HeT

Error correcting

• [1 0 1]T is the syndrome of the errors• [1 0 0 1 1 0] has an error in the second

entry• The corrected message is [1 1 0 1 1 0]

Summary

• Codewords of length 6• 3-bit messages• At most one error• Use generator matrix to encode and parity

check matrix to decode

Hamming code (1950)

• {0000000, 1101001, 0101010, 1000011, 1001100, 0100101, 1100110, 0001111, 1110000, 0011001, 1011010, 0110011, 0111100, 1010101, 0010110, 1111111} the (7,4) Hamming code

An application of linear codes

• In 1971 Mariner 9 transmitted pictures of Mars back to earth

• The distance between Mars and earth is 84 million miles

• The transmitter on Mariner 9 had only 20 watts

Why eigenvectors

• Diagonalization• Write A = PDP-1 where D is a diagonal

matrix Then An = PDnP-1 (used in Markov chains)

• Alternatively use geometry

EIGENVECTORS

11

411

3113

11

211

3113

4 and 2 are called eigenvalues

11

11and are called eigenvectors

What are they for?

1

33113

11

3113

211

3113

08

11

2211

4

Suppose

1

12

11

13 Then

We reduce matrix multiplication to scalar multiplication.

Geometric meaning

13

01

3113

11

411

3113

Finding eigenvectors using geometry

Finding eigenvectorsusing geometry

13

01

3113

11

411

3113

31

10

3113

22

11

3113

13

01

3113

44

11

3113

31

10

3113

2

21

13113

Using eigenvectors as coordinates

3113

13

08

into maps

3113

21

44

)2(214

maps into

Using eigenvectors as coordinates

Eigenvectors as geometry

• To find eigenvectors is to find new coordinates

• To find new coordinates is to simplify computation

• Linear algebra is by no means abstract

Two recent reports

• www.ed.gov/MathPanel• www.reform.co.uk/documents/The%20val

ue%20of%20mathematics.pdf

To teach mathematics is to teach skills and rigour

END

pengyee.lee@nie.edu.sg