02-24-03

# 02-24-03 - Lecture 10 Andrei Antonenko 1 Meaning of linear...

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Unformatted text preview: Lecture 10 Andrei Antonenko February 24, 2003 1 Meaning of linear dependence and independence On the last lecture we stated the result that if the system of vectors is linearly dependent, then at least one vector of the. system can be expressed as a linear combination of others. We gave an example how to do it. Now we’ll give an example when it is not possible to express any vector as a linear combination of others. Example 1.1. Let u 1 = 1 , u 2 = 1 , u 3 = 1 Here u 1 ,u 2 and u 3 are linearly independent and none of these vectors can be expressed as a linear combination of other 2 vectors. For example, for u 1 there are no real a and b such that u 1 = 1 = au 2 + bu 3 = a 1 + b 1 2 Spanning sets Definition 2.1. Let V be a vector space. Vectors v 1 ,v 2 ,...,v n are called a spanning set of V if every element of V is a linear combination of v 1 ,v 2 ,...,v n . In this case the space V is called a span of these vectors and it is denoted by V = h v 1 ,v 2 ,...,v n i Example 2.2. Consider the vector space R 3 . Then vectors v 1 = (1 , , 0) , v 2 = (0 , 1 , 0) , and v 3 = (0 , , 1) form a spanning set of R 3 , since if u ∈ V equals to ( a,b,c ) , then u = av 1 + bv 2 + cv 3 . Example 2.3. Consider the vector space R 3 . Then vectors v 1 = (1 , 1 , 1) , v 2 = (1 , 1 , 0) , and v 3 = (1 , , 0) form a spanning set of R 3 , since if u ∈ V equals to ( a,b,c ) , then u = cv 1 + ( b- c ) v 2 + ( a- b ) v 3 . For example, (4 , 6 , 1) = 1(1 , 1 , 1) + 5(1 , 1 , 0)- 2(1 , , 0) . 1 Example 2.4. Consider the vector space P ( t ) . Then vectors 1 ,t,t 2 ,t 3 ,... are a spanning set of P ( t ) since it is clear that every polynomial can be expressed as a linear combination of these vectors....
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02-24-03 - Lecture 10 Andrei Antonenko 1 Meaning of linear...

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