Question

Let V be a vector space and v, 02, .., Un be e and v1, v2,... n be vectors in V. Prove that if v1, U2,..,Un are linearly dependent, then one of the vectors vi must be a linear combination of the other vectors.

Answer 1

To prove that if v1, v2, ... vn are linearly dependent, then one of the vectors vi must be a linear combination of the other vectors, we can use the definition of linear dependence and assume that v1, v2, ... vn are linearly dependent. Then, we can show that one of the vectors can be expressed as a linear combination of the other vectors.

Step-by-step explanation:

In order to prove that if v1, v2, ... vn are linearly dependent, then one of the vectors vi must be a linear combination of the other vectors, we need to use the definition of linear dependence. A set of vectors is linearly dependent if there exist scalars c1, c2,..., cn, not all zero, such that c1v1 + c2v2 +... + cnvn = 0.

Assume that v1, v2, ... vn are linearly dependent. This means there exist scalars c1, c2,..., cn, not all zero, such that c1v1 + c2v2 +... + cnvn = 0. Without loss of generality, assume that c1 ≠ 0, then we can write v1 = (-c2/c1)v2 +... + (cn/c1)vn.

Therefore, we have shown that one of the vectors is a linear combination of the other vectors.