Question

29. A subset B of a vector space V over F is called a basis of V, if:
(A) B is linearly independent set only
(B) B spans V only
(C) B is linearly independent set and spanning set
(D) None of these​

Answer 1

Answer:

Theorem 2.3. A subset B of a vector space V over a field F is a basis if and only if it is a maximal linearly independent subset of V . ... Let F denote the collection of all linearly independent subsets containing S. The collection F is non-empty because S ∈ F. Let L be the chain in F.

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

A subset B of a vector space V over F is called a basis of V, if

(A) B is linearly independent set only

(B) B spans V only

(C) B is linearly independent set and spanning set

(D) None of these

CONCEPT TO BE IMPLEMENTED

First we recall the definition of Basis of a Vector Space

Basis :

Let V be a vector space over a field F . A set S of vectors in V is said to be a basis of V if

• S is linearly independent in V

• S generates V

Example :

The set E = { (1,0,0),(0,1,0),(0,0,1) } is the standard basis of $\sf{ \mathbb{R} {}^{3} }$

EVALUATION

From above we can conclude that

A subset B of a vector space V over F is called a basis of V, if

(C) B is linearly independent set and spanning set

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