True or False: If the dimension of a vector space is n, then there exists a linearly independent set of V that contains (n -1) vectors.

Answer 1

If the vector space has dimension n then any basis for the space must contain exactly n linearly independent vectors.

However, if the vector space has dimension n then it is possible to have a set of n-1 vectors in the space that is linearly independent,

For example, R³ has dimension 3 but the set {<1,0,0>, <0,1,0>} is a linearly independent set of vectors in R³, but is not a basis of R³.

So, the answer to your question is TRUE.