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 9 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, 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 What we have learnt • defined/studied a mathematical system  called a real vector space, and • noted some of its properties. • Observe that the only real vector space having a finite number of vectors in it is the vector space whose only vector is , • for if v 6 = is in a vector space V , then the following exercise Show that if v 6 = and a v = b v , then a = b , a v 6 = b v , where a and b are distinct real numbers, so V has infinitely many vectors in it. What is the next? • We shall show that most vector spaces V studied here have a set composed of a finite number of vectors that completely describe V ; that is, we can write every vector in V as a linear combination of the vectors in this set. • We shall also see that there is more than one such set describing V . We now turn to a formulation of these ideas. LINEAR INDEPENDENCE 4 DEFINITION  The vectors v 1 , v 2 ,..., v k in a vector space V are said to span V if every vector in V is a linear combination of v 1 , v 2 ,..., v k . Moreover, if S = { v 1 , v 2 ,..., v k } , then we also say that then we also say • that the set S spans V , or • that v 1 , v 2 ,..., v k spans V , or • that V is spanned by (or generated by ) S , or • in the language of the subspace (cf. Lecture 7/8), span S = V . The main goal is to use a set of vectors to generate a vector space . LINEAR INDEPENDENCE 5 The procedure to check if the vectors v 1 , v 2 ,..., v k span the vector space V is as follows. Step 1. Choose an arbitrary vector v in V Step 2. Determine if v is a linear combination of the given vectors. If it is, then the given vectors span V . If it is not, they do not span V . Remark: In Step 2, we investigate the consistency of a linear system , but this time for a right side that represents an arbitrary vector in a vector space V . LINEAR INDEPENDENCE 6 Example 1 Let V be the vector space R 3 and let v 1 = (1 , 2 , 1) , v 2 = (1 , , 2) and v 3 = (1 , 1 , 0) . Do v 1 , v 2 and v 3 span V ? LINEAR INDEPENDENCE 7 Solution Step 1. Let v = ( a,b,c ) be any vector in R 3 , where a,b and c are arbitrary real numbers. Step 2. We must find out whether there are constants c 1 ,c 2 and c 3 such that c 1 v 1 + c 2 v 2 + c 3 v 3 = v . This leads to the linear system (verify) c 1 + c 2 + c 3 = a 2 c 1 + c 3 = b c 1 + 2 c 2 = c....
View
Full
Document
This note was uploaded on 05/18/2010 for the course MATHEMATIC MAT2310 taught by Professor Dr.jeffchakfuwong during the Spring '06 term at CUHK.
 Spring '06
 Dr.JeffChakFuWONG
 Algebra, Vector Space

Click to edit the document details