Let us consider the following system of linear equations:

x1 −2x2 −x3 + 3x4 = 0,

−2x1 + 4x2 + 5x3 −5x4 = 3,

3x1 −6x2 −6x3 + 8x4 = 2.

do the following.

(a) (2pts) is this linear system a square, underdetermined, or overdetermined system? why?

(b) (2pts) rewrite the linear system in the matrix-vector form.

(c) (2pts) write the augmented matrix of the linear system.

(d) (10pts) use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. show your work as how we did in the classroom!
1

(e) (2pts) what are the pivot positions? which columns are the pivot columns?

(f) (2pts) among the four variables {x1,x2,x3,x4}, which are the basic variables and which are the free variables?

(g) (2pts) decide whether or not the system is consistent. if it is consistent, does the system have a unique solution or infinitely many solutions? ( read theorem 2 on page 21)

(h) (6pts) continue row reduction to obtain the reduced echelon form of the augmented matrix.

(i) (2pts) is the statement "if a linear system has at least one free variable, then the system must have infinitely many solutions" true or false? why?