UNIT20 - Unit 20 Independence and Basis in n The idea of...

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Unit 20 Independence and Basis in < n The idea of dimension is fairly intuitive. Consider a vector in < n ,( a 1 ,a 2 3 , ..., a n ). Each of the n components are independent i.e. choosing a single compo- nent in no way eFects the choice for the other components. ±or this reason we say that < n has n degrees of freedom or equivalently has dimension n , i.e. < 3 has dimension 3, < 2 has dimension 2 etc. Consider now the “subspace” (soon to be de²ned) of < 3 consisting of all vectors ( x, y, z )w ith x + y + z = 0. We may now only choose 2 components freely and these determine the third. So this subspace is said to have dimen- sion 2. Defnition 22.1. By a linear combination of the vectors ~v 1 ,~v 2 , .., ~v k in < n we mean an expression of the form c 1 1 + c 2 2 + ... + c k k when each c i is a scalar. Example 1. Let v 1 =( 2 , 0 , 1), 2 =(1 , 1 , 2) and 3 =(4 , 2 , 3). Express 3 as a linear combination of 1 and 2 . Solution: We must ²nd scalars c 1 and c 2 so that 3 = c 1 1 + c 2 2 we set (4 , 2 , 3 )= c 1 ( 2 , 0 , 1) + c 2 (1 , 1 , 2) (4 , 2 , 3 )=( 2 c 1 , 0 ,c 1 )+( c 2 , c 2 , 2 c 2 ) (4 , 2 , 3 2 c 1 + c 2 , c 2 1 2 c 2 ) 1
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Equating components we have 4 2 3 = 2 c 1 + c 2 c 2 c 1 2 c 2 Solving, we have c 1 = 1and c 2 =2. We now give two defnitions For the same thing. Each defnition is (oF course) equivalent to the other, but sometimes the use oF one defnition is more suited to a given task than the other. Defnition 22.2. A set oF vectors in < n is said to be linearly dependent iF one vector in the set can be expressed as a linear combination oF the others.
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This note was uploaded on 04/15/2010 for the course MATH 20C taught by Professor Lit during the Spring '10 term at UCLA.

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UNIT20 - Unit 20 Independence and Basis in n The idea of...

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