Consider the 5 imes 6 matrix
P=([2,1,1,3,0,4],[4,2,4,4,1,5],[2,1,3,1,0,4],[6,3,4,8,1,9],[0,0,3,-3,0,0])
with columns labelled C_(1),C_(2),dots,C_(6). Answer the questions and justify your answers.
(a) What is the rank of P ?
(b) Which are the basic columns of P ?
(c) Express the non-basic columns of P as linear combinations of the basic columns.
(d) Suppose that P=(A|B) is the augmented matrix for a 5 imes 5 system of linear equations Ax=B.
What are all solutions?
(e) If P is as in (3d), what are all solutions to the homogeneous system Ax=0 ?
3.Consider the 5 6 matrix
2 1 1 3 0 4 4 2 4 4 1 5 2 1 3 1 0 4 6 3 4 8 1 6 0 0 3 3 0 0
P=
with columns labelled Ci,C2,...,C6. Answer the questions and justify your answers.
(a) What is the rank of P? (b) Which are the basic columns of P?
(c) Express the non-basic columns of P as linear combinations of the basic columns.
(d) Suppose that P = (A|B) is the augmented matrix for a 5 5 system of linear equations AX = B. What are all solutions?
(e) If P is as in (3d), what are all solutions to the homogeneous system AX = 0?