Lecture_7_1 - Lecture Note 7-1 Oct 24 Dr Jeff Chak-Fu WONG...

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Unformatted text preview: Lecture Note 7-1: Oct 24 - Oct 27, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 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 8. Orthonormal Bases in R n 9. Orthogonal Complements REAL VECTOR SPACES 2 C OORDINATES AND C HANGE OF B ASIS COORDINATES AND CHANGE OF BASIS 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 . However, in the discussion of this lecture 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 . 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 AND CHANGE OF BASIS 4 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 AND CHANGE OF BASIS 5 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 AND CHANGE OF BASIS 6 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 ] . COORDINATES AND CHANGE OF BASIS 7 Transforming the matrix in (1) to reduced row echelon form, we obtain the solution (verify) c 1 = 3 , c 2 =- 1 , c 3 =- 2 , c 4 = 1 , so the coordinate vector of v with respect to the basis S is [ v ] S =        3- 1- 2 1        ....
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Lecture_7_1 - Lecture Note 7-1 Oct 24 Dr Jeff Chak-Fu WONG...

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