First, we will transform the matrix to the reduced echelon form [1 0 0 1 1 0 -4 0 -8 1 -1 0 8 -2 16] rightarrow [1 0 0 1 1 0 -4 0 1 1 -1 0 8 -2 -2] this reduced row echelon form of the augmented matrix corresponds to the system the leading enters in the matrix have been highlighted in blue a leading entry on the (I,j) position indicates that the j-th unknown will be determined using the i-th equation those columns in the coefficient part of the matrix that don’t contain leading entries, corresponds to unknown that will be arbitrary the system has implemented many solutions x_1 = -2x_4-2x_5 x_2 = 1x_4 + 2x_5 x_3 = 2x_5 x_4 = arbitrary = c_4 x_5 = arbitrary = c_5 the solution can be written in the vector form c_4[-2 1 0 1 0]+c_5[-2 2 2 0 1] therefore the null space has a basis formed by the set set { [-2 1 0 1 0],[-2 2 2 0 1]}

First, we will transform the matrix to the reduced echelon form [1 0 0 1 1 0 -4 0 -8 1 -1 0 8 -2 16] rightarrow [1 0 0 1 1 0 -4 0 1 1 -1 0 8 -2 -2] this reduced row echelon form of the augmented matrix corresponds to the system the leading enters in the matrix have been highlighted in blue a leading entry on the (I,j) position indicates that the j-th unknown will be determined using the i-th equation those columns in the coefficient part of the matrix that don’t contain leading entries, corresponds to unknown that will be arbitrary the system has implemented many solutions x_1 = -2x_4-2x_5 x_2 = 1x_4 + 2x_5 x_3 = 2x_5 x_4 = arbitrary = c_4 x_5 = arbitrary = c_5 the solution can be written in the vector form c_4[-2 1 0 1 0]+c_5[-2 2 2 0 1] therefore the null space has a basis formed by the set set { [-2 1 0 1 0],[-2 2 2 0 1]}