Question

Let v_(1),1<=i<=n, be vectors in a vector space V. Show that a vector vinSpan (v_(1),v_(2),cdots, v_(n)) can be written uniquely as a linear combination of v_(i,),1<=i<=n, if nd only if v_(i,1)<=i<=n, are linearly independent.

Answer 1

A vector can be uniquely written as a linear combination of a set of vectors if and only if those vectors are linearly independent. Uniqueness implies linear independence, whereas linearly independent vectors ensure that no vector in the span is the result of more than one set of coefficients.

Step-by-step explanation:

The question asks us to demonstrate that a vector v in the span of a set of vectors 1 ≤ i ≤ n can be written as a unique linear combination of these vectors if and only if the set {v_i} is linearly independent. This is a core concept in linear algebra, relating to vector spaces and linear combinations of vectors.

Firstly, if a vector can be written as a unique linear combination of vectors {v_i}, that implies no other combination of scalar multiples of these vectors will give the same vector v. This uniqueness inherently means that the only way for a linear combination of the {v_i} vectors to result in the zero vector is if all scalar coefficients are zero. This is the definition of linear independence.

Conversely, if the vectors {v_i} are linearly independent, no vector in the span can be formed by more than one distinct set of coefficients in a linear combination of the {v_i} vectors. If there were different sets of coefficients that could generate the same vector v in the span, it would imply a linear dependency among the vectors {v_i}, which contradicts our assumption of linear independence. Therefore, a vector can be uniquely written only when the {v_i} are linearly independent.