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Chapter 5-BASIS AND DIMENSION LECTURE 7PowerPoint Presentation

Chapter 5-BASIS AND DIMENSION LECTURE 7

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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,...vmV are said to be linearly dependent over K, or simply dependent, if there exist scalars a1,...,amK, 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.

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

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.

Theorem 5.4:

Suppose the set (v1, v2, ..., vn) generates a vector space V. If (w1, ... Wm) is linearly independent, then mn 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.

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.

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.

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 Wn. In particular if dim W=n, then W=V.

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 (UW)

Note that if V is the direct sum of U and W, i.e. V = U ⊕W, then dimV = dim U + dim W.

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.

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.

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, aiK

Theorem 5.12:

Let V be an n-dimensional vector space over a field K. Then V and Kn are isomorphic.

VKn .

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.

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