LINEAR INDEPENDENCE Definition An indexed set of vectors

LINEAR INDEPENDENCE § Definition: An indexed set of vectors {v 1, …, vp} in is said to be linearly independent if the vector equation has only the trivial solution. The set {v 1, …, vp} is said to be linearly dependent if there exist weights c 1, …, cp, not all zero, such that ----(1) © 2012 Pearson Education, Inc. Slide 1. 7 - 1

LINEAR INDEPENDENCE § Equation (1) is called a linear dependence relation among v 1, …, vp when the weights are not all zero. § An indexed set is linearly dependent if and only if it is not linearly independent. § Example 1: Let © 2012 Pearson Education, Inc. , , and . Slide 1. 7 - 2

a. Determine if the set {v 1, v 2, v 3} is linearly independent. b. If possible, find a linear dependence relation among v 1, v 2, and v 3. § Solution: We must determine if there is a nontrivial solution of the following equation. © 2012 Pearson Education, Inc. Slide 1. 7 - 3

LINEAR INDEPENDENCE § Row operations on the associated augmented matrix show that. § x 1 and x 2 are basic variables, and x 3 is free. § Each nonzero value of x 3 determines a nontrivial solution of (1). § Hence, v 1, v 2, v 3 are linearly dependent. © 2012 Pearson Education, Inc. Slide 1. 7 - 4

LINEAR INDEPENDENCE b. To find a linear dependence relation among v 1, v 2, and v 3, row reduce the augmented matrix and write the new system: § § § Thus, , , and x 3 is free. Choose any nonzero value for x 3—say, Then and. © 2012 Pearson Education, Inc. . Slide 1. 7 - 5

LINEAR INDEPENDENCE § Substitute these values into equation (1) and obtain the equation below. § This is one (out of infinitely many) possible linear dependence relations among v 1, v 2, and v 3. © 2012 Pearson Education, Inc. Slide 1. 7 - 6