Lecture 19 - Feb 23

Lecture 19 - Feb 23 - Monday February 23 Lecture 19 :...

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Monday February 23 Lecture 19 : Coordinate of a vector with respect to a basis. . (Refers to section 4.4) Expectations: 1. Recall: When expressing a vector v in V as a linear combination of the vectors in a basis, the coefficients are unique. 2. Give a graphic representation of vectors for different bases in R 2 and R 3 . 3. Define and find the coordinates of v with respect to a given basis of a vector space V . 19.1 Recall The unique representation theorem . If V is a vector space and B = { v 1 , v 2 , .... , v d } forms a basis for V then for every v in V there is a unique set of coefficients α 1 , α 2, ...., α d in R such that v = α 1 v 1 + α 2 v 2 + . ... + α d v d . Proof (Outline) ± Suppose v = α 1 v 1 + α 2 v 2 + .... + α d v d = β 1 v 1 + β 2 v 2 + . ... + β d v d . ± Then 0 = ( α 1 β 1 ) v 1 + ( α 2 β 2 ) v 2 + . ... + ( α d − β d ) v d ± Since v 1 , v 2 , .... , v d are linearly independent then α i β i = 0 for all i. 19.2 Definition If v is a vector in the vector space V which has basis B = { v 1 , v 2 , .... , v d } (which we will call the B -basis of V , so that we can distinguish it from other bases) then the unique coefficients α 1 , α 2, ...., α d used to express v
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Lecture 19 - Feb 23 - Monday February 23 Lecture 19 :...

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