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Lecture_9 - Lecture Note 9 Dr Jeff Chak-Fu WONG Department...

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Lecture Note 9 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1
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R EAL V ECTOR S PACES 1. Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis and Dimension 5. Coordinates and Change of Basis 6. Homogeneous Systems 7. The Rank of a Matrix and Applications R EAL V ECTOR S PACES 2
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L INEAR I NDEPENDENCE L INEAR I NDEPENDENCE 3
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What we have learnt defined/studied a mathematical system - called a real vector space, and noted some of its properties. Observe that the only real vector space having a finite number of vectors in it is the vector space whose only vector is 0 , for if v 6 = 0 is in a vector space V , then the following exercise Show that if v 6 = 0 and a v = b v , then a = b , a v 6 = b v , where a and b are distinct real numbers, so V has infinitely many vectors in it. What is the next? We shall show that most vector spaces V studied here have a set composed of a finite number of vectors that completely describe V ; that is, we can write every vector in V as a linear combination of the vectors in this set. We shall also see that there is more than one such set describing V . We now turn to a formulation of these ideas. L INEAR I NDEPENDENCE 4
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DEFINITION - The vectors v 1 , v 2 , . . . , v k in a vector space V are said to span V if every vector in V is a linear combination of v 1 , v 2 , . . . , v k . Moreover, if S = { v 1 , v 2 , . . . , v k } , then we also say that then we also say that the set S spans V , or that v 1 , v 2 , . . . , v k spans V , or that V is spanned by (or generated by ) S , or in the language of the subspace (cf. Lecture 7/8), span S = V . The main goal is to use a set of vectors to generate a vector space . L INEAR I NDEPENDENCE 5
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The procedure to check if the vectors v 1 , v 2 , . . . , v k span the vector space V is as follows. Step 1. Choose an arbitrary vector v in V Step 2. Determine if v is a linear combination of the given vectors. If it is, then the given vectors span V . If it is not, they do not span V . Remark: In Step 2, we investigate the consistency of a linear system , but this time for a right side that represents an arbitrary vector in a vector space V . L INEAR I NDEPENDENCE 6
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Example 1 Let V be the vector space R 3 and let v 1 = (1 , 2 , 1) , v 2 = (1 , 0 , 2) and v 3 = (1 , 1 , 0) . Do v 1 , v 2 and v 3 span V ? L INEAR I NDEPENDENCE 7
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Solution Step 1. Let v = ( a, b, c ) be any vector in R 3 , where a, b and c are arbitrary real numbers. Step 2. We must find out whether there are constants c 1 , c 2 and c 3 such that c 1 v 1 + c 2 v 2 + c 3 v 3 = v . This leads to the linear system (verify) c 1 + c 2 + c 3 = a 2 c 1 + c 3 = b c 1 + 2 c 2 = c. A solution is (verify) c 1 = - 2 a + 2 b + c 3 , c 2 = a - b + c 3 , c 3 = 4 a - b - 2 c 3 . Since we have obtained a solution for every choice of a, b and c , we conclude that v 1 , v 2 , v 3 span R 3 . This is equivalent to saying that span { v 1 , v 2 , v 3 } = R 3 .
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