If the columns of a matrix
with
rows and
columns do not span Rⁿ, then there exists a vector
in Rⁿ such that
does not have a solution.

Question

If the columns of a matrix with rows and columns do not span Rⁿ, then there exists a vector in Rⁿ such that does not have a solution.

Quizplus · Accepted Answer

This is a statement of the rank-nullity theorem, which states that the rank of a matrix plus the dimension of its null space is equal to the number of columns in the matrix. If the columns of the matrix do not span the vector space, then its rank is less than the number of columns, which means its null space has a non-trivial dimension. This implies the existence of a non-zero vector in the null space, which does not have a solution in the column space (since it is orthogonal to all the columns).

If the Columns of a Matrix