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Unformatted text preview: Chapter 04.02 Vectors After reading this chapter, you should be able to: 1. define a vector, 2. add and subtract vectors, 3. find linear combinations of vectors and their relationship to a set of equations, 4. explain what it means to have a linearly independent set of vectors, and 5. find the rank of a set of vectors. What is a vector? A vector is a collection of numbers in a definite order. If it is a collection of n numbers, it is called a ndimensional vector. So the vector A given by = n a a a A 2 1 is a ndimensional column vector with n components, n a a a ,......, , 2 1 . The above is a column vector. A row vector ] [ B is of the form ] ,...., , [ 2 1 n b b b B = where B is a n dimensional row vector with n components n b b b ,...., , 2 1 . Example 1 Give an example of a 3dimensional column vector. Solution Assume a point in space is given by its ) , , ( z y x coordinates. Then if the value of 5 , 2 , 3 = = = z y x , the column vector corresponding to the location of the points is 04.02.1 04.02.2 Chapter 04.02 = 5 2 3 z y x . When are two vectors equal? Two vectors A and B are equal if they are of the same dimension and if their corresponding components are equal. Given = n a a a A 2 1 and = n b b b B 2 1 then B A = if n i b a i i ,......, 2 , 1 , = = . Example 2 What are the values of the unknown components in B if = 1 4 3 2 A and = 4 1 4 3 b b B and B A = . Solution 1 , 2 4 1 = = b b How do you add two vectors? Two vectors can be added only if they are of the same dimension and the addition is given by Vectors 04.02.3 + = + n n b b b a a a B A 2 1 2 1 ] [ ] [ + + + = n n b a b a b a 2 2 1 1 Example 3 Add the two vectors = 1 4 3 2 A and  = 7 3 2 5 B Solution  + = + 7 3 2 5 1 4 3 2 B A + + + = 7 1 3 4 2 3 5 2 = 8 7 1 7 04.02.4 Chapter 04.02 Example 4 A store sells three brands of tires: Tirestone, Michigan and Copper. In quarter 1, the sales are given by the column vector = 6 5 25 1 A where the rows represent the three brands of tires sold Tirestone, Michigan and Copper respectively. In quarter 2, the sales are given by...
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 Spring '12
 MUNK

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