ch4 - CHAPTER 4 VECTOR SPACES The treatment of vector...

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Section 4.1 187 CHAPTER 4 VECTOR SPACES The treatment of vector spaces in this chapter is very concrete. Prior to the final section of the chapter, almost all of the vector spaces appearing in examples and problems are subspaces of Cartesian coordinate spaces of n -tuples of real numbers. The main motivation throughout is the fact that the solution space of a homogeneous linear system Ax = 0 is precisely such a "concrete" vector space. SECTION 4.1 THE VECTOR SPACE R 3 Here the fundamental concepts of vectors, linear independence, and vector spaces are introduced in the context of the familiar 2-dimensional coordinate plane R 2 and 3-space R 3 . The concept of a subspace of a vector space is illustrated, the proper nontrivial subspaces of R 3 being simply lines and planes through the origin. 1. ( 2 , 5 ,4 ) ( 1 ,2 ,3 ) ( 1 , 7 ,1 ) 5 1 −= ab 2 2 ( 2 , 5 1 4 , 1 0 ,8 1 5 , 8 1 ) 3 4 3(2,5, 4) 4(1, 2, 3) (6,15, 12) (4, 8, 12) (2,23,0) += −+−−= −− = 2. ( 1,0,2) (3,4, 5) ( 4, 4,7) 81 9 = = 2 2( 1,0,2) (3,4, 5) ( 2,0,4) (3,4, 5) (1,4, 1) 3 4 3( 1,0,2) 4(3,4, 5) ( 3,0,6) (12,16, 20) ( 15, 16,26) +=− + −=− + = 3. (2 3 5 ) (5 3 7 ) 3 6 12 189 3 21 + −+ = + = = i j k 22 ( 2 3 5 ) ( 5 3 7 ) ( 461 0 )( 537 ) 933 34 3 ( 2 35 ) 4 ( 5 37 ) (6 9 15 ) (20 12 28 ) 14 21 43 −+ + +− =− + + + = + +− + + + = + ij k ijk k k i j k i j k 4. (2 ) ( 3 ) 2 2 3 17 − = −−− = − + = ij j k 2 2(2 ) ( 3 ) (4 2 ) ( 3 ) 4 3 3 4 3(2 ) 4( 3 ) (6 3 ) (4 12 ) 6 7 12 + = + + = + a b i j jk i j i i j j k k
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188 Chapter 4 5. 3 2 , = vu so the vectors u and v are linearly dependent. 6. (0,2) (3,0) (3 ,2 ) ab a b b a += + = = uv 0 implies a = b = 0, so the vectors u and v are linearly independent. 7. (2,2) (2, 2) (2 2 ,2 2 ) a b + = + = 0 implies a = b = 0, so the vectors u and v are linearly independent. 8. , =− so the vectors u and v are linearly dependent. In each of Problems 9–14, we set up and solve (as in Example 2 of this section) the system 11 1 22 2 w a w b  = =   w to find the coefficient values a and b such that , =+ wu v 9. 1 3, 2 so 3 2 23 0 a b = == = + v 10. 32 0 2, 3 so 2 3 43 1 a b = = v 11. 52 1 1, 2 73 1 a b = = v 12. 42 2 5 so 3 5 2 a b = = + −− v 13. 5 2, 2 so 2 3 54 2 a b = = v 14. 56 5 7, 5 so 7 5 24 6 a b = = + v In Problems 15–18, we calculate the determinant uvw so as to determine (using Theorem 4) whether the three vectors u , v , and w are linearly dependent (det = 0) or linearly independent (det 0).
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Section 4.1 189 15. 358 14 3 0 26 4 −= −− so the three vectors are linearly dependent. 16. 524 23 5 0 45 7 = so the three vectors are linearly dependent.
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ch4 - CHAPTER 4 VECTOR SPACES The treatment of vector...

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