63 problems 1 show that 886 is a linear combination

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.
Intermediate Algebra Within Reach
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.
Intermediate Algebra Within Reach
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.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture