Youtor
161 Field axioms
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http://youtube.com/watch?v=D-TLk0rriZ8
An introduction to the axioms of the real numbers: properties that describe what is interesting about the real numbers, but that cannot be deduced from other properties. We then go on to derive theorems based on these axioms. We then discuss how complex numbers are also a field, but that matrix-matrix multiplication does not satisfy the axioms of a field. • Course web site: https://ece.uwaterloo.ca/~ne112/Lectu... • 0:00 Introduction • 0:48 The useful properties of the real numbers • 2:40 Generalizing these properties as fields • 3:17 Another field: the rational numbers • 3:55 Another field: p + q√2 • 5:44 Another field: the integers 0, ..., p - 1 where p is prime • 7:23 Examples of non-fields • 9:12 Theorem: 0 + α = α • 9:48 Theorem: 0 is unique • 10:30 Theorem: The reciprocal of the reciprocal of α is α • 11:34 The benefit: all these theorems apply to all the described fields • 12:05 Defining other operations: subtraction, division and integer exponentiation • 14:20 Theorem: the reciprocal of a product is the product of the reciprocals • 15:25 Theorem: 1/(1/α + 1/β) = αβ/(α + β) • 16:47 Claim (to be shown later): the complex numbers form a field • 17:14 How simplifications are useful • 17:59 Warning: you assume αβ = βα, but this doesn't work with matrices • 18:39 A review of the axioms • 19:23 Important take-aways from this topic • 20:18 Summary and conclusions • This is part of the course NE 112 Linear algebra for nanotechnology engineering offered by the Department of Electrical and Computer Engineering within the Faculty of Engineering at the University of Waterloo (https://www.uwaterloo.ca/).
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