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Basis and Dimension

Definition of basis:

(A) A basis for a vector space is a sequence of vectors that is linearly Independent and that spans the space.

(B) A set S={u1 , u2 , ….. , un}of vectors is a basis of V if it has the following two properties: (i) S is linearly independent (ii) S spans V.

(C) A set S={u1 , u2 , ….. , un}of vectors is a basis of V if every v∈V can be written uniquely as a linear combination of the basis vectors.

Example: This is a basis for R2

 

bd1

It is linearly independent.

 

bd2bd3

By solving equation (1) and (2) we get

bd4

Definition of standard basis: For any Rn

bd5

is the standard basis. We denote these vectors

bd6

Theorem: In any vector space, a subset is a basis if and only if each vector in the space can be expressed as a linear combination of elements of the subset in one and only one way.

 

Proof: A sequence is a basis if and only if its vectors form a set that spans and that is linearly independent. A subset is a spanning set if and only if each vector in the space is a linear combination of elements of that subset in at least one way. Thus we need only show that a spanning subset is linearly independent if and only if every vector in the space is a linear combination of elements from the subset in at most one way.

Consider two expressions of a vector as a linear combination of the members of the subset. Rearrange the two sums, and if necessary add some

bd7

terms, so that the two sums combine the same β ̅ ’s in the same order:

bd8

Nowbd9

holds if and only if

bd10

holds. So, asserting that each coefficient in the lower equation is zero is the same

thing as asserting that ci = di for each i, that is, that every vector is expressible

as a linear combination of the β ̅ ’s in a unique way. (proved)

 

 

 

Definition of finite dimension:

 

(A) A vector space is finite-dimensional if it has a basis with only finitely many vectors.

(B) A vector space V is said to be of finite dimension n or n-dimensional, written div V=n

if V has a basis with n elements.

The vector space {0} is defined to have dimension 0.

Suppose a vector space V does not have a finite basis. Then V is said to be of infinite dimension or to be finite dimensional.

 

 

 

 

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