The set 2 3 3 1 8 9 0 4 12 10 2 0 4 5 1 3 is linearly indepe

The set {[2 3 -3 1], [-8 -9 0 -4], [12 10 2 0], [-4 -5 1 -3]} is linearly independent. Does the system 2x_1 - 8x_2 + 12x_3 - 4x_4 = 53 3x_1 - 9x_2 + 10x_3 - 5x_4 = -71 -3x_1 + 2x_3 + x_4 = 109 x_1 - 4x_2 - 3x_4 = pi^2 have a solution? If yes, is it unique?

Solution

Since given set of linearly independent vectors is columns of coefficient matrix A of system Ax=b.

Therefore, rank A = 4 = rank (A l b )

So, system is consistent and so it has a solution.

Since rank A = no. Of unknowns

Hence given system has a unique solution.