sense making in linear algebra lee peng yee bangkok 10-07-2008
DESCRIPTION
Contents Vector spaces and bases Matrices Eigenvalues and eigenvectorsTRANSCRIPT
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Sense making in linear algebra
Lee Peng YeeBangkok10-07-2008
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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
011101110
100010001
G
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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
011011011101110
100010001
011
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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
100010001
011101110
H
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Decoding
000
011011
100010001
011101110
101
011001
100010001
011101110
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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
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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 An = PDnP-1 (used in Markov chains)
• Alternatively use geometry
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EIGENVECTORS
11
411
3113
11
211
3113
4 and 2 are called eigenvalues
11
11and are called eigenvectors
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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.
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Geometric meaning
13
01
3113
11
411
3113
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Finding eigenvectors using geometry
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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
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Using eigenvectors as coordinates
3113
13
08
into maps
3113
21
44
)2(214
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.pdf
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To teach mathematics is to teach skills and rigour