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Sense making in linear algebra

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

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Sense making in linear algebra

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  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

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