UNIT13 - Unit 13 Euclidean n-Space and Linear Equations...

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Unit 13 Euclidean n-Space and Linear Equations Euclidean n-Space: < n Much of this half of the course will involve objects called vectors. In order to introduce this subject we begin with some deFnitions and notation: Defnition 14.1. By < we will mean the set of all real numbers. By < n we will mean the set of all n-tuples: ~u =( u 1 ,u 2 3 , ..., u n - 1 n ) where for each i, u i is a real number. ~u is called an n-vector or just a vector , and u i is called the i th component of ~u . We say that two vectors ~u and ~v are equal , written ~u = if each pair of corresponding components are equal. i.e. u i = v i for every i . for example, ~u =(1 , 1 , 2) ∈< 3 , 2 , 1) 3 and ~w , 1 , 2) 3 .W eh a v e ~u 6 = ~v since u 2 6 = v 2 ,and ~u = . A number will be refered to here as a scalar .So1 ,0 , π ,e and -5 are all exam- ples of scalars. So we now have deFned two kinds of objects, vectors and scalars. The question now arises as to how we may combine these two things ( can we add them together, multiply them together etc.). We now address this question. Defnition 14.2. We deFne the operation of scalar multiplication of a vec- tor: Let ~u u 1 2 3 , ..., u n - 1 n ) n and c any scalar then we deFne: c~u cu 1 ,cu 2 3 , ..., cu n - 1 n ) 1
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Example 1. We have 3(1 , 2 , 3) = (3 , 6 , 9) 2(1 , 3 , 4 , 0) = (2 , 6 , 8 , 0) and π (1 , 1 , 2 , 2 , 3) = ( π,π, 2 π, 2 3 π ) Defnition 14.3. By the zero vector , denoted ~ 0 ∈< n , we will mean the n-vector whose components are all zero. Example 2. ~ 0=(0 , 0) in < 2 ~ , 0 , 0) in < 3 and ~ , 0 , 0 , 0) in < 4 Defnition 14.4. We defne the operation oF vector addition : IF ~u =( u 1 ,u 2 3 , ..., u n - 1 n )and ~v v 1 ,v 2 3 , ..., v n - 1 n ) are both From < n then we may defne the sum oF the two vectors as: ~u + u 1 + v 1 2 + v 2 3 + v 3 , ..., u n - 1 + v n - 1 n + v n ) Example 3.
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UNIT13 - Unit 13 Euclidean n-Space and Linear Equations...

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