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 .
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Prof. Dr. Zafer ASLAN
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”.
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;
Otherwise the vectors are said to be linearly independent over K, or simply independent.
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:
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.
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.
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.
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.
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
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.
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.
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
Let V be an n-dimensional vector space over a field K. Then V and Kn are isomorphic.
Chapter 6: Linear Mappings
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.