Lecture 13-Fill in - Math 415 Lecture 13 Basis and...

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Math 415 - Lecture 13 Basis and Dimension Monday September 26th 2016 Textbook reading: Chapter 2.3 Suggested practice exercises: Chapter 2.3 Exercise 1, 2, 3, 5, 6, 9, 11, 16, 19, 20, 22, 27. Khan Academy video: Introduction to Linear Independence, More on linear inde- pendence, Span and Linear Independence Example, Basis of a Subspace Strang lecture: Independence, Basis, and Dimension 1 Review Vectors v 1 , . . . , v p are linearly Dependent if x 1 v 1 + x 2 v 2 + · · · + x p v p = 0 , and not all the coefficients are zero. The columns of A are linearly IN dependent ⇐⇒ each column of A contains a pivot ⇐⇒ there are no free variables for A x = 0 . Are the vectors 1 1 1 , 1 2 3 , - 1 1 3 independent? 1
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Any set of 11 vectors in R 10 is linearly dependent. Why? Definition 1. In a list of vectors ( v 1 , . . . , v p ) in a vector space V we call v k redundant if v k is a linear combination of the previous vectors. In this case Span( v 1 , v 2 , . . . , v k - 1 , v k ) = Span( v 1 , v 2 , . . . , v k - 1 ), i.e., you can delete the redundant vector and get the same span.
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