Lecture_12 - Lecture Note 12 Dr. Jeff Chak-Fu WONG...

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Unformatted text preview: Lecture Note 12 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1 R EAL V ECTOR S PACES 1. Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis and Dimension 5. Homogeneous Systems 6. The Rank of a Matrix and Applications 7. Coordinates and Change of Basis REAL VECTOR SPACES 2 C OORDINATES COORDINATES 3 • If V is an n-dimensional vector space, we know that V has a basis S with n vectors in it; • so far we have not paid much attention to the order of the vectors in S . Here, we speak of an ordered basis S = { v 1 , v 2 ,..., v n } for V ; thus S = { v 2 , v 1 ,..., v n } is a different ordered basis for V . COORDINATES 4 Ordered basis: A set of vectors S = { v 1 , v 2 , ··· , v n } in a vector space V is called an ordered basis for V provided S is a basis for V and if we reorder the vectors in S , the new ordering of the vectors in S is considered a different basis for V . COORDINATES 5 An important reason for specifying a basis S for a vector space V is to impose a “ coordinate system " on V . • We shall show that if S contains n vectors, then the coordinate system will make V act like R n . • If V is already R n itself, then S will determine a coordinate system that gives a new view of V . COORDINATES 6 • – If S = { v 1 , v 2 ,..., v n } is an ordered basis for the n-dimensional vector space V , – then every vector v in V can be uniquely expressed in the form v = c 1 v 1 + c 2 v 2 + ··· + c n v n , where c 1 ,c 2 ,...,c n are real numbers. • We shall refer to [ v ] S = c 1 c 2 . . . c n as the coordinate vector of v with respect to the ordered basis S . • The entries of [ v ] S are called the coordinate of v with respect to S . COORDINATES 7 • Observe that the coordinate vector [ v ] S depends on the order in which the vectors in S are listed; a change in the order of this listing may change the coordinates of v with respect to S . • All bases considered in this section are assumed to be ordered bases. COORDINATES 8 Example 1 Let S = { v 1 , v 2 , v 3 , v 4 } be a basis for R 4 , where v 1 = (1 , 1 , , 0) , v 2 = (2 , , 1 , 0) , v 3 = (0 , 1 , 2 ,- 1) , v 4 = (0 , 1 ,- 1 , 0) . If v = (1 , 2 ,- 6 , 2) , compute [ v ] S . Solution To find [ v ] S we need to compute constants c 1 ,c 2 ,c 3 and c 4 such that c 1 v 1 + c 2 v 2 + c 3 v 3 + c 4 v 4 = v , which is just a linear combination problem. The previous equation leads to the linear system whose augmented matrix is (verify) 1 2 1 1 1 1 2 1 2- 1- 6- 1 2 , (1) or equivalently, [ v T 1 v T 2 v T 3 v T 4 | v T ] ....
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Lecture_12 - Lecture Note 12 Dr. Jeff Chak-Fu WONG...

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