Wild Linear Algebra 20: Bases of polynomial spaces
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This lecture studies spaces of polynomials from a linear algebra point of view. We are especially interested in useful bases of a four dimensional space like P^3: polynomials of degree three or less. We introduce the standard (or power) basis, also the modified Factorial basis. Translations of the corresponding functions yield linear transformations, giving Taylor bases and a purely algebraic definition of the derivative. We see that some basic calculus ideas are really algebraic in nature, not requiring `real numbers', limits or slopes of tangents. This course in Linear Algebra is given by N J Wildberger. CONTENT SUMMARY: pg 1: @00:08 Map of a space of polynomials to a space of vectors; Definition: linear/vector space; course distinction @03:27 ; pg 2: @04:46 Definition: An ordered basis of a linear space; Example1 ; pg 3: @07:03 Example2 (basis of a vector space); Example3 (basis of a polynomial space); Definition: Dimension of a linear/vector space; examples; pg 4: @09:25 The space of polynomials is richer than the isomorphic space of vectors; translating polynomials @09:50 ; degree of the polynomial is preserved in translation; pg 5: @13:20 Study01 of translation by 3 (see previous page); A linear transformation @16:15 ; pg 6: @16:48 study01 continued; image and kernel of polynomial of degree 3 @17:16; pg 7: @19:04 the derivative of a function appears in translation; pg 8: @22:49 Definition: The derivative of a polynomial; calculus via linear algebra @23:00; pg8_Theorem; factorial notation; pg 9: @28:05 Calculus as algebra; pg9_Theorem (product rule); proof (Leibniz mentioned); pg 10: @33:46 Translating a polynomial and obtaining the derivatives; Taylor series mentioned; pg 11: @37:58 Importance of various bases; standard basis; factorial basis; Example ; coefficient vectors of a polynomial with respect to a basis; pg 12: @42:10 Theorem (Basis isomorphism correspondence); The standard vector space of column vectors @45:30 ; pg 13: @46:22 The derivative as a linear transformation; remark about formulas in calculus and combinatorics @50:46 ; pg 14: @51:13 Another basis; the standard basis moved over by 3; example; pg 15: @54:32 Change of basis matrix; How to get this matrix! @54:56 ; pg 16: @57:16 Exercises 20.1-3; closing remarks @58:27; (THANKS to EmptySpaceEnterprise)
Comments
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This is an awesome video :D Im definatly subbing, i will try to visit your channel regularly from now on :D if you can do visit my channel! That would be awesome!! :D Until next time and keep up the great videos! Peace
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@16:60, t14=-27.
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Excellent job. Thanks for taking the time to do this.
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So cool!
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Awesome lecture, very helpful and intuitive.
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there's another honorable lecutre from... DrChrisTisDell...
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I think it is important to realize that what you are doing here using rational numbers can be extended to elements of a general field, for example, the complex numbers, and the calculus becomes "complex analysis".
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Thank you very much. Really love the teaching style!!
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A good lecture!!
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This is great, tnxxx<3
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Beautiful demonstation of Grassmann's analysis brought up to date and with your own insights. Love it ,prof!
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njwildberger, MIT, nptelhrd, patrickJMT, and khanacademy. What a beautiful time to learn about science and math!
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I love the way teaching , please do the whole course of Linear Algebra, thank you again :)
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Thank you
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