Lecturenotes28February2011

Lecturenotes28February2011 - Section 4.3 Linear...

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Unformatted text preview: Section 4.3 Linear Independence Linear indepencency of vectors will be used to define basis of a vector space that we will see in section 4.4 and to determine the dimension of a space that we will see in section 4.5. Definition: A non-empty set of vectors S = { v 1 ,v 2 ,...,v s } is said to be linearly in- dependent if the only scalars c 1 ,c 2 ,...,c s that satisfy the equation c 1 v 1 + c 2 v 2 + ... + c s v s = are c 1 = 0 ,c 2 = 0 ,...,c s = 0 . If there are scalars, not all zero, that satisfy this equa- tion, then the set is said to be linearly dependent. Note: Linearly independency and dependency is defined for finite sets. For infinite sets we call them linearly independent if every finite subset is linearly independent. Example: 1. { ,v 1 ,v 2 ,...,v s } is linearly dependent as 1 · 0 + 0 v 1 + ... + 0 v s = 0 2. v 1 ,v 2 be two vectors, consider c 1 v 1 + c 2 v 2 = 0 v 1 ,v 2 linearly dependent means v 1 =- c 2 c 1 v 2 or v 2 =- c 1 c 2 v 1 . v 1 ,v 2 linearly independent if v 1 ,v 2 are not scalar multiple of each other. 3. (1 , , 0) , (0 , 1 , 0) and (0 , , 1) are linearly independent. 4. In fact, { e 1 ,e 2 ,...,e n } is a linearly independent set in R n ....
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Lecturenotes28February2011 - Section 4.3 Linear...

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