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lect136_16_w09

# lect136_16_w09 - Monday February 9 Lecture 16 Linear...

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Monday February 9 Lecture 16: Linear independence (Refers to section 4.2) Expectations : 1. Define a linearly independent set. 2. Recognize that subsets of linearly independent sets are linearly independent. 3. Characterize a linearly independent set as one being a set where no vector is a linear combination of the others. 4. Recognize that a matrix A x = 0 only has the trivial solution iff its columns are linearly independent. 5. Determine when a subset of R n is linearly independent. 16.1 Definition – Let { v 1 , v 2, .... , v k } be a spanning family for a vector space V . Then for every vector v in V there exists scalars α 1 , α 2 , ..... , α k such that v = α 1 v 1 + α 2 v 2 + ..... + α k v k . If for every vectors v this set of scalars α 1 , α 2 , ..., α k associated to v in this way is unique then we say that this spanning family satisfies the unique representation property . 16.1.1 Example – The spanning family {(1, 0), (0,1)} = { e 1 , e 2 } satisfies the unique representation property since for an arbitrary vector v = ( a , b ) in R 2 , a and b are the unique scalars associated to v so that v = a e 1 + b e 2 . 16.1.2 Example – The family of vectors { v 1 , v 2 , v 3 } = { x 3 + x 2 + 1, 2 x 3 + 1/2, 3 x 3 x 2 } in P 3 (the family of all polynomials of degree 3 or less including the 0-polynomial) does not satisfy the unique representation property since the vector 0 (the 0- polynomial) has two ways of representing it as a linear combination of v 1 , v 2 and v 3 : 0 = 1( x 3 + x 2 + 1) + (–2) (2 x 3 + ½) + 1(3 x 3 x 2 ) = v 1 + (–2) v 2 + 1 v 3 and 0 = 0( x 3 + x 2 + 1) + 0(2 x 3 + ½) + 0(3 x 3 x 2 ) = 0 v 1 + 0 v 2 + 0 v 3

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So we have two sets of scalars {1, –2, 1} and {0, 0, 0} associated to the 0 -vector in this way. s 16.2 Definition – Let v 1 , v 2, .... , v k be k vectors in a vector space V . The vectors v 1 , v 2, .... , v k are said to be linearly independent if the ONLY way that α 1 v 1 + α 2 v 2 + ..... + α k v k = 0
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lect136_16_w09 - Monday February 9 Lecture 16 Linear...

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