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mat 2310 0607_2_note4_1

mat 2310 0607_2_note4_1 - Lecture Note 4-1 Feb 5 2007 Dr...

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Lecture Note 4-1: Feb 5, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 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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DEFINITION-The vectors v 1 , v 2 , . . . , v k in a vector space V are said to be linearly dependent if there exist constants c 1 , c 2 , . . . , c k , not all zero, such that c 1 v 1 + c 2 v 2 + · · · + c k v k = 0 . (1) Otherwise, v 1 , v 2 , . . . , v k are called linearly independent . That is, v 1 , v 2 , . . . , v k are linearly independent if whether c 1 v 1 + c 2 v 2 + · · · + c k v k = 0 , we must have c 1 = c 2 = · · · = c k = 0 . That is, the only linear combination of v 1 , v 2 , . . . , v k that yields the zero vector is that in which all the coefficients are zero. If S = { v 1 , v 2 , . . . , v k } , then we also say that the set S is linearly dependent or linearly independent if the vectors have the corresponding property defined previously. It should be emphasized that for any vectors { v 1 , v 2 , . . . , v k } , Equation (1) always holds if we choose all the scalars c 1 , c 2 , . . . , c k equal to zero. The important point in this definition is whether or not it is possible to satisfy (1) with at least one of the scalars different from zero. L INEAR I NDEPENDENCE 4
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The procedure to determine if the vectors { v 1 , v 2 , . . . , v k } are linearly dependent or linearly independent is as follows. Step 1. Form Equation (1), which leads to a homogeneous system. Step 2. 1. If the homogeneous system obtained in Step 1 has only a trivial solution , then the given vectors are linearly independent ; 2. if it has a nontrivial solution , then the vectors are linearly dependent . L INEAR I NDEPENDENCE 5
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Example 1 Determine whether the vectors - 1 1 0 0 and - 2 0 1 1 found in Example 40 (cf. Lecture Note 3) as spanning the solution space of A x = 0 are linearly dependent or linearly independent. L INEAR I NDEPENDENCE 6
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Solution Forming Equation (1) c 1 - 1 1 0 0 + c 2 - 2 0 1 1 = 0 0 0 0 , we obtain the homogeneous system - c 1 - 2 c 2 = 0 c 1 + 0 c 2 = 0 0 c 1 + c 2 = 0 0 c 1 + c 2 = 0 , whose only solution is c 1 = c 2 = 0 . Hence the given vectors are linearly independent. L INEAR I NDEPENDENCE 7
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Example 2 Are the vectors v 1 = (1 , 0 , 1 , 2) , v 2 = (0 , 1 , 1 , 2) and v 3 = (1 , 1 , 1 , 3) in R 4 linearly dependent or linearly independent? Solution We form Equation (1) c 1 v 1 + c 2 v 2 + c 3 v 3 = 0 and solve for c 1 , c 2 and c 3 . The resulting homogeneous system is (verify) c 1 + c 3 = 0 c 2 + c 3 = 0 c 1 + c 2 + c 3 = 0 2 c 1 + 2 c 2 + 3 c 3 = 0 , which has as its only solution c 1 = c 2 = c 3 = 0 (verify), showing that the given vectors are linearly independent.
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