1 / 14

Chapter 5-BASIS AND DIMENSION LECTURE 7

Chapter 5-BASIS AND DIMENSION LECTURE 7. Prof. Dr. Zafer ASLAN. BASIS AND DIMENSION. INTRODUCTION Some of the fundamental results proven in this chapter are: i) The “dimension” of a vector space is well defined. ii) If V has dimension n over K, then V is “isomorphic” to K n .

denna
Download Presentation

Chapter 5-BASIS AND DIMENSION LECTURE 7

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 5-BASIS AND DIMENSION LECTURE 7 Prof. Dr. Zafer ASLAN

  2. BASIS AND DIMENSION INTRODUCTION Some of the fundamental results proven in this chapter are: i) The “dimension” of a vector space is well defined. ii) If V has dimension n over K, then V is “isomorphic” to Kn. iii) A system of linear equations has a solution if and only if the coefficient and augmented matrices have the same “rank”. LINEAR DEPENDENCE Definition: Let V be a vector spaceover a field K. The vectors v1,...vmV are said to be linearly dependent over K, or simply dependent, if there exist scalars a1,...,amK, not all of then 0, such that; a1v1+a2v2+...+amvm=0 Otherwise the vectors are said to be linearly independent over K, or simply independent.

  3. BASIS AND DIMENSION Theorem 5.1: The nonzero rows of a matrix in echelon form are linearly independent. For more than one vector, the concept of dependence an be defined equivalently as follows: The vectors v1,...vm are linearly dependent if and only if one of them is a linear combination of the others. For suppose, say vi is a linear combination of the others: vi= a1v1+a2v2+...+ai-1vi-1+ai+1vi+1+amvm

  4. BASIS AND DIMENSION Definition: A vector space V is said to be of finite dimension n or to be n-dimensional, written dimV =n, if there exists linearly independent vectors e1, e2, ..., en which span V. The sequence (e1, e2, ..., en) is then called a basis of V. The above definition of dimension is well defined in viiew of the following theorem: Theorem 5.3: Let V be a finite dimensional vector space. Then every basis of V has the same number of elements. The vector space (0) is defined to have dimension 0. (In a certain sense this agrees with the above definition since by definition, Øis independent and generates (0)). When a vector space is not of finite dimension, it is said to be of infinite dimension.

  5. BASIS AND DIMENSION Theorem 5.4: Suppose the set (v1, v2, ..., vn) generates a vector space V. If (w1, ... Wm) is linearly independent, then mn and V is generated by a set of the form: (w1, ...wm, vi1,...vin-m) Thus, in particular, any n+1 or more vectors in V are linearly dependent.

  6. BASIS AND DIMENSION Theorem 5.5: Suppose S generates V and (v1, v2, ..., vn)is a maximal independent subset of S. Then (v1, ... vm) is a basis of V. The main relationship between the dimension of a vector space and its independent subsets is contained in the next theorem.

  7. BASIS AND DIMENSION Theorem 5.6: Let V be of finite dimension n. Then: i) Any set of n+1 or more vectors is linearly dependent, ii) Any linearly independent set is part of a basis, i.e. Can be extended to a basis. iii) A linearly independent set with n elements is a basis.

  8. BASIS AND DIMENSION DIMENSION AND SUBSPACES The following theorems give basic relationships between the dimension of a vector space and the dimension of a subspace. Theorem 5.7: Let W be a subspace of an n-dimensional vector space V. Then dim Wn. In particular if dim W=n, then W=V.

  9. BASIS AND DIMENSION Theorem 5.8: Let U and W be a finite-dimensional subspaces of a vector space V. Then U+W has finite dimension and dim (U+V) = dimU + dimW – dim (UW) Note that if V is the direct sum of U and W, i.e. V = U ⊕W, then dimV = dim U + dim W.

  10. BASIS AND DIMENSION RANK OF A MATRIX Let A be an arbitrary m+n matrix over a field K. Recall that the row space of A is the subspace of Km generated by its rows, and he column space of A is the subspace of Km generated by its columns. The dimensions of the row space and of the column space of A are called, respectively, the row rank and the column rank of A. Theorem 5.9: The row rank and the column rank of the matrix are equal. Definition: The rank of the matrix A, written rank (A), is the common value of its row rank and column rank. Thus the rank of a matrix gives the maximum number of independent rows, and also the maximum number of independent columns.

  11. BASIS AND DIMENSION APPLICATIONS TO LINEAR EQUATIONS Consider a system of m linear equations in n unknowns x1, ..., xn over a field K. The equivalent matrix equation: AX = B (A,B) = Thus the system AX=B has a solution if and only if the column vector B is a linear combination of the columns of the matrix A, i.e. Belongs to the column space of A.

  12. BASIS AND DIMENSION Theorem 5.10: The system of linear equations AX=B has a solution if and only if the coefficient matrix A and the augmented matrix (A,B) have the same rank. Theorem 5.11: The dimension of the solution space W of the homogeneous system of linear equations AX = 0 is n-r where n is the number of unknowns and r is the rank of the coefficient matrix A. COORDINATES Let (ei, ... , en) be a basis of a n – dimensional vector space V over a field K, and let v be any vector in V. Since (ei) generates V, v is a linear combination of the ei: v = a1e1+a2e2+...+anen, aiK

  13. BASIS AND DIMENSION Theorem 5.12: Let V be an n-dimensional vector space over a field K. Then V and Kn are isomorphic. VKn .

  14. Next Lecture (Week 8) Chapter 6: Linear Mappings Reference Seymour LIPSCHUTZ, (1987): Schaum’s Outline of Theory and Problems of LINEAR ALGEBRA, SI (Metric) Edition, ISBN: 0-07-099012-3, pp. 334, McGraw – Hill Book Co., Singapore.

More Related