Sense making in linear algebra

1. Sense making in linear algebra Lee Peng Yee Bangkok 10-07-2008

2. Historical events • Geometry went algebraic after Felix Klein • Algebra turned abstract • Linear algebra came from geometry

3. Contents • Vector spaces and bases • Matrices • Eigenvalues and eigenvectors

4. Two questions • Why linear algebra or motivation • Why eigenvalues and eigenvectors

5. Why linear algebra • Linear systems • Geometric transformations • Markov chains • Lately linear codes

6. LINEAR CODES • Message  encode  transmit  received  decode  detect error  correct error  final message • Concepts used: vector space/linear space, basis, matrices, matrix multiplication

7. 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

8. 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

9. Generator matrix

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

11. Encoding

12. 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?

13. Parity check matrix

14. Decoding

15. 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

16. 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]

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

18. Hamming code (1950) • {0000000, 1101001, 0101010, 1000011, 1001100, 0100101, 1100110, 0001111, 1110000, 0011001, 1011010, 0110011, 0111100, 1010101, 0010110, 1111111} the (7,4) Hamming code

19. 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

20. Why eigenvectors • Diagonalization • Write A = PDP-1 where D is a diagonal matrix Then An = PDnP-1 (used in Markov chains) • Alternatively use geometry

21. EIGENVECTORS 4 and 2 are called eigenvalues and are called eigenvectors

22. What are they for? Suppose Then We reduce matrix multiplication to scalar multiplication.

23. Geometric meaning

24. Finding eigenvectors using geometry

25. Finding eigenvectorsusing geometry

26. Using eigenvectors as coordinates into maps maps into

27. Using eigenvectors as coordinates

28. 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

29. Two recent reports • www.ed.gov/MathPanel • www.reform.co.uk/documents/The%20value%20of%20mathematics.pdf

30. To teach mathematics is to teach skills and rigour

31. END pengyee.lee@nie.edu.sg