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Unformatted text preview: Lecture Note 41: Feb 5, 2007 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff ChakFu WONG 1 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 REAL VECTOR SPACES 2 L INEAR I NDEPENDENCE LINEAR INDEPENDENCE 3 DEFINITIONThe 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 = . (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 = , 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. LINEAR INDEPENDENCE 4 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 . LINEAR INDEPENDENCE 5 Example 1 Determine whether the vectors  1 1 and  2 1 1 found in Example 40 (cf. Lecture Note 3) as spanning the solution space of A x = are linearly dependent or linearly independent. LINEAR INDEPENDENCE 6 Solution Forming Equation (1) c 1  1 1 + c 2  2 1 1 = , we obtain the homogeneous system c 1 2 c 2 = c 1 + c 2 = c 1 + c 2 = c 1 + c 2 = , whose only solution is c 1 = c 2 = 0 . Hence the given vectors are linearly independent. LINEAR INDEPENDENCE 7 Example 2 Are the vectors v 1 = (1 , , 1 , 2) , v 2 = (0 , 1 , 1 , 2) and v 3 = (1 , 1 , 1 , 3) in R 4 linearly dependent or linearly independent?...
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This note was uploaded on 06/05/2011 for the course MATH 3333 taught by Professor Jeffwong during the Spring '11 term at CUHK.
 Spring '11
 JeffWong
 Linear Algebra, Algebra

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