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Unformatted text preview: Review II: Nov 28  Dec 1, 2006 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff ChakFu WONG 1 R EAL V ECTOR S PACES 1. Basis and Dimension 2. Homogeneous Systems 3. The Rank of a Matrix and Applications 4. Coordinates and Change of Basis 5. Orthonormal Bases in R n 6. Orthogonal Complements REAL VECTOR SPACES 2 List of Nonsingular Equivalence The following statements are equivalent for an n × n matrix A . 1. A is nonsingular. 2. x = is the only solution to A x = 3. A is row equivalent to I n . 4. The linear system A x = b has a unique solution for every n × 1 matrix b . 5. det( A ) 6 = 0 . 6. A has rank n . 7. A has nullity 0. 8. The rows of A form a linearly independent set of n vectors in R n . 9. The columns of A form a linearly independent set of n vectors in R n . REAL VECTOR SPACES 3 4. B ASIS AND D IMENSION L ECTURE N OTE 5 Basis DEFINITIONThe vectors v 1 , v 2 ,..., v k in a vector space V are said to form a basis for V if 1. (a) v 1 , v 2 ,..., v k span V , and 2. (b) v 1 , v 2 ,..., v k are linearly independent. Theorem 0.1 If S = { v 1 , v 2 ,..., v n } is a basis for a vector space V , then every vector in V can be written in one and only one way as a linear combination of the vectors in S . Theorem 0.2 Let S = { v 1 , v 2 ,..., v n } be a set of nonzero vectors in a vector space V and let W = span S . Then some subset of S is a basis for W . 4. BASIS AND DIMENSION  LECTURE NOTE 5 4 Let V = R m and let S = { v 1 , v 2 ,..., v n } be a set of nonzero vector in V . The procedure for finding a subset of S that is a basis for W = span S is as follows. Step 1. Form Equation (3) c 1 v 1 + c 2 v 2 + ··· + c n v n = . Step 2. Construct the augmented matrix associated with the homogeneous system of Equation (3) and transform it to reduced row echelon form. Step 3. The vectors corresponding to the columns containing the leading 1s form a basis for W = span S . 4. BASIS AND DIMENSION  LECTURE NOTE 5 5 Theorem 0.3 If S = { v 1 , v 2 ,..., v n } is a basis for a vector space V and T = { w 1 , w 2 ,..., w r } is a linearly independent set of vectors in V , then r ≤ n . Corollary 0.1 If S = { v 1 , v 2 ,..., v n } and T = { w 1 , w 2 ,..., w r } are bases for a vector space, then n = m . 4. BASIS AND DIMENSION  LECTURE NOTE 5 6 Dimension DEFINITIONThe dimension of a nonzero vector space V is the number of vectors in a basis for V . We often write dim V . Since the set { } is linearly dependent, it is natural to say that the vector space { } has the dimension zero . Theorem 0.4 If S is a linearly independent set of vectors in a finitedimensional vector space V , then there is a basis T for V , which contains S ....
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This note was uploaded on 10/15/2009 for the course MATHEMATIC MAT2310B taught by Professor Jeffwong during the Spring '09 term at CUHK.
 Spring '09
 JeffWong
 Linear Algebra, Algebra

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