psLinAlgebra1Qu - such that u = .v ) c) for n > 1, ( v 1...

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Fall 2007 ARE211 Problem Set #06 First Lin Algebra Problem Set Due date: Oct 16 (3) Suppose you know that the angle between two vectors is as given below. What do you know about the sign of the inner product of the two vectors? a) 180 b) 53 c) 320 d) 90 (4) Using the deFnition of linear independency,i.e., a set of vectors { v 1 , ...v k ..., v m } is a linear independent set if for all t R m , m k =1 t k v k = 0 implies t = 0, prove the following properties: (Note: once you have shown a property, you can use it to show the following ones) a) A singleton vector is a linear independent set if and only if it is not the zero vector. b) Two nonzero vectors are linearly independent if and only if they are not colinear (or proportional, i.e. for two vectors ( u, v ) there exists
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Unformatted text preview: such that u = .v ) c) for n > 1, ( v 1 , ..., v n ) is a linear dependent set if and only if one of the vector in the set v i is a linear combination of the other n-1 vectors. d)If ( v 1 , ..., v n ) is a linear independent set, and y is a dierent vector, ( v 1 , ..., v n , y ) is linearly dependent if and only if y is a linear combination of v 1 , ..., v n e) If ( v 1 , ..., v n ) is a linear independent set, then any subset of this set (such as v 1 , ..., v i , with i < n ) is also linear independent. 2 (5) Suppose we are in R 2 , what is the maximum number n of vectors that will make a linear independent set? Take show that any other vector can be expressed as a linear combination of these n linearly independent vectors. What about in R 3 ?...
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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psLinAlgebra1Qu - such that u = .v ) c) for n > 1, ( v 1...

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