chapter3.page14

Chapter3.page14 - only if x = sy for some s 6 = 0 • n nonzero vectors are linearly independent if one can be rep-resented as linear combination

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14 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES Notice: - Press [Shift][/]to get the derivative operator and press [Ctrl][I] to get the antiderivative operator. - To get dx ( t ) simplify you type dx ( t ) and press [Shift][Ctrl][.] and type the key word simplify in the place holder before . - To execute symbolically ( operator), just press [Ctrl][.] 3. Linearly independency 3.1. Linearly independency of vectors. Let x 1 ,x 2 , ··· ,x n be n vectors, C 1 ,C 2 , ··· ,C n are n scalars(numbers), the expression C 1 x 1 + C 2 x 2 + ··· + C n x n is called a linear combination of vectors x 1 ,x 2 , ··· ,x n . Definition 3.1 . n vectors x 1 ,x 2 , ··· ,x n is linearly independent if C 1 x 1 + C 2 x 2 + ··· + C n x n = 0 leads to C 1 = 0 ,C 2 = 0 , ··· ,C n = 0 . A set of vectors are linearly dependent if they are not linearly in- dependent. If 0 is one of x 1 ,x 2 , ··· ,x n , then they linearly dependent. Two nonzero vectors x and y are linearly dependent if and
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Unformatted text preview: only if x = sy for some s 6 = 0 . • n nonzero vectors are linearly independent if one can be rep-resented as linear combination of the others. • Any three or more 2-dimensional vectors (vectors with two entries) are linear dependent. • Any four or more 3-dimensional(vectors with three entries) vectors are linear dependent. To determine if a given set of vectors are linearly independent, create a matrix so that the row of the matrix are given vectors. Using Mathcad function rref( ) to find the reduced echelon form of the matrix, if the result contains one or more rows that are entirely zero the vectors are linearly dependent, otherwise the vectors are linearly independent....
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

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