Is it possible to find a 4x5 rank 3 matrix B such that the g

Is it possible to find a 4x5, rank 3 matrix B such that the given xi all satisfy BXi=0

Solution

The question is incomplete as the Xis have not been stated.

The rank–nullity theorem states that the sum of the rank and the nullity of a matrix is equal to the number of columns of the matrix. Since B is a 4X5 matrix of rank 3, its nullity is 5 -3 = 2. Thus, the equation BX = 0 has 2 linearly independent solutions.. All the other solutions of the equation BX = 0 are linear combinations of these two linearly independent solutions. Thus, if the given Xi vectors are more than 2 which are also linearly independent, then the equation B Xi = 0 cannot be satisfied for i 3.Then it is not possible to find the matrix B satisfying the given requirements. However, if i 2, then a matrix B satisfying the given conditions can be found.