# 63 problems 1 show that 886 is a linear combination

• Notes
• GrandIceBaboon7537
• 26
• 100% (2) 2 out of 2 people found this document helpful

This preview shows page 8 - 10 out of 26 pages.

##### We have textbook solutions for you!
The document you are viewing contains questions related to this textbook. The document you are viewing contains questions related to this textbook.
Chapter 5 / Exercise 98
Intermediate Algebra Within Reach
Larson Expert Verified
6.3 Problems 1. Show that (8,8,6) is a linear combination of (1,2,1) and (2,0,1). 2. Show that (7,8,9) is a linear combination of (1,2,3) and (4,5,6). 3. Show that (5,2,4) is not a linear combination of (1,2,1) and (2,0,1). 4. Show that (6,9,1) is not a linear combination of (1,2,3) and (4,5,6). 5. Is (5,3,2) in Span { (1,1,0), (1,0,1) } ? 6. Is (4,3,2) in Span { (1,1,0), (1,0,1) } ? 7. Show that Span { (1,1,0), (1,0,1), (2,1,1) } = Span { (1,1,0), (1,0,1) } . 8. Show that Span { (1,2), (2,1) } = Span { (1,0), (0,1) } . 9. Let W = { ( a, b, a + 2 b ) a, b real numbers } . Find a spanning set for W . 6.4 Linear dependence and independence In the previous section we saw that it may be possible to delete a vector from a spanning set and have the remaining vectors span the same space as the original set of vectors. In this section we will consider finding a spanning set containing a minimal number of vectors. A nonempty set of vectors { in a vector space V is said to be linearly independent if the only solution to the vector equation = = r 1 v 1 + r 2 v 2 r n v n = 0 is the trivial solution r 1 = r 2 " " } , , , { n 2 1 v v v " " 0 k r r n = 0. The set is said to be linearly dependent if there is a solution with some . Example Determine whether the set S = { (1,2, 3), (4,5,6), (7,8,9) } is linearly dependent or linearly independent. 8
##### We have textbook solutions for you!
The document you are viewing contains questions related to this textbook. The document you are viewing contains questions related to this textbook.
Chapter 5 / Exercise 98
Intermediate Algebra Within Reach
Larson Expert Verified
Solution Set r 1 (1,2,3) + r 2 (4,5,6) + r 3 (7,8,9) = (0,0,0) and solve for r 1 , r 2 and r 3. We get ( r 1 + 4 r 2 + 7 r 3 , 2 r 1 + 5 r 2 + 8 r 3 , 3 r 1 + 6 r 2 + 9 r 3 ) = (0,0,0). Equating the corresponding components on each side of the equation produces the following system of three equations: r 1 + 4 r 2 + 7 r 3 = 0 2 r 1 + 5 r 2 + 8 r 3 = 0 3 r 1 + 6 r 2 + 9 r 3 = 0. Use matrix methods to solve this homogeneous system of linear equations. 1 4 7 0 1 4 7 0 1 4 7 0 1 0 1 0 2 5 8 0 0 3 6 0 0 1 2 0 0 1 2 0 3 6 9 0 0 6 12 0 0 0 0 0 0 0 0 0 . There is a one- parameter family of solutions ( t , –2 t , t ), and so the set S is linearly dependent. Thus, for example, if we take t = 1 we get 1(1, 2, 3) + (–2)(4, 5, 6) + 1(7, 8, 9) = (0, 0, 0). Example Determine whether the set S = { (1,1,0), (1,0,1), (0,1,1) } is linearly dependent or linearly independent. Solution Set r 1 (1,1,0) + r 2 (1,0,1) + r 3 (0,1,1) = (0,0,0) and solve for r 1 , r 2 and r 3. We get ( r 1 + r 2 , r 1 + r 3 , r 2 + r 3 ) = (0,0,0). Equating the corresponding components on each side of the equation gives the following system of three homogeneous linear equations: r 1 + r 2 = 0 r 1 + r 3 = 0 r 2 + r 3 = 0. Use matrix equations to solve this system. 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 2 1 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 The only solution is r 1 = r 2 = r 3 = 0, so the set S is linearly independent. 0 v Consider a set consisting of only one vector. Let S = { v }. If , the set S is linearly independent since the only solution to the vector equation r v = 0 is r = 0. On the other hand, if v = 0 , the set is linearly dependent as the equation r v = 0 has a nonzero solution since, for example, if r = 1 and v = 0 we get r v = 1( 0 ) = 0 . Theorem Let S be a set of 2 or more vectors in a vector space V . If S is a linearly dependent set then some vector in the set can be written as a linear combination of the other vectors in the set. Conversely, if some vector from S can be written as a linear combination of the other vectors of S, then S is a linearly dependent set.
• • • 