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Sense making in linear algebra Lee Peng Yee Bangkok 10-07-2008
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Historical events Geometry went algebraic after Felix Klein Algebra turned abstract Linear algebra came from geometry
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Contents Vector spaces and bases Matrices Eigenvalues and eigenvectors
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Two questions Why linear algebra or motivation Why eigenvalues and eigenvectors
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Why linear algebra Linear systems Geometric transformations Markov chains Lately linear codes
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LINEAR CODES Message encode transmit received decode detect error correct error final message Concepts used: vector space/linear space, basis, matrices, matrix multiplication
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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
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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
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Generator matrix
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Message [110] is a 3-bit message We turn it into a codeword (encode) Transmit the codeword Then decode
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Encoding
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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?
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Parity check matrix
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Decoding
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Error detecting If Hx T = [0 0 0] T then x is a codeword If Hx T = [1 0 1] T then en error is detected If x is a codeword, r is received word, and e is error then Hr T = Hx T + He T = He T
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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]
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Summary Codewords of length 6 3-bit messages At most one error Use generator matrix to encode and parity check matrix to decode
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Hamming code (1950) {0000000, 1101001, 0101010, 1000011, 1001100, 0100101, 1100110, 0001111, 1110000, 0011001, 1011010, 0110011, 0111100, 1010101, 0010110, 1111111} the (7,4) Hamming code
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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
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Why eigenvectors Diagonalization Write A = PDP -1 where D is a diagonal matrix Then A n = PD n P -1 (used in Markov chains) Alternatively use geometry
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EIGENVECTORS 4 and 2 are called eigenvalues and are called eigenvectors
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What are they for? Suppose Then We reduce matrix multiplication to scalar multiplication.
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Geometric meaning
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Finding eigenvectors using geometry
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Using eigenvectors as coordinates into maps into
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Using eigenvectors as coordinates
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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
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Two recent reports www.ed.gov/MathPanel www.reform.co.uk/documents/The%20val ue%20of%20mathematics.pdfwww.reform.co.uk/documents/The%20val ue%20of%20mathematics.pdf
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To teach mathematics is to teach skills and rigour
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END pengyee.lee@nie.edu.sg
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