This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture Note 71: Oct 24  Oct 27, 2006 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2006 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. 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 ndimensional 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 ndimensional 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 ....
View
Full
Document
 Spring '09
 JeffWong
 Linear Algebra, Algebra

Click to edit the document details