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Lecture 17 - Feb 11

# Lecture 17 - Feb 11 - Wednesday February 11 Lecture 17...

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Wednesday February 11 Lecture 17 : Examples on linear independence (Refers to section 4.2) Expectations : 1. Define a linearly independent set. 2. Determine when a subset of R n is linearly independent. 17.1 Theorem Let S = { v 1 , v 2 , …, v k } be a set of k distinct non-zero vectors in R n . If k > n then the set S cannot be linearly independent. Proof: Let A be the matrix whose columns are v 1 , v 2 , …, v k . Then A is an n by k matrix. If k > n then A has more columns then rows and so A RREF must point to at least one free variable in the system A x = 0 . Hence A x = 0 has infinitely many solutions and so has non-trivial solutions. Thus any set with more vectors then the dimension of R n is not linearly independent. Thus if S = { v 1 , v 2 , …, v k } is linearly independent in R n then k n . 17.2 Example Show that the set { e x , x 2 } is linearly independent in the vector space of all real-valued continuous functions on R .

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Lecture 17 - Feb 11 - Wednesday February 11 Lecture 17...

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