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Choose the false statement: The set of vectors is said to be linearly independent if

Choose the false statement: The set of vectors is said to be linearly independent if Correct Answer Rank of matrix with these vector as column is less than number of given vectors.

Concept:

Linearly independent:

  • A set of vectors {v1, v2,…, vp} in a vector space V is said to be linearly independent if the vector equation c1v1 + c2v2 +…+ cpvp = 0 has only one trivial solution c1 = 0, c2 = 0,…, cp = 0
  • The matrix with these vector as column has non-zero determinant.
  • Its every finite subset is linearly independent.
  • Number of given vectors are same as the rank of matrix.

Explanation:

From the above discussion, we can conclude that, 

A set of vectors is to be linearly independent, a rank of a matrix with these vectors should be equal to the number of given vectors.

Hence, option 4 is incorrect.

Additional InformationLinearly dependent:

1. The set is said to be linearly dependent if there exists weights c1, c2,…, cp not all 0, such that c1v1 + c2v2 +…+ cpvp = 0

2. The rank of a matrix with these vectors as columns is less than the number of given vectors.

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