1. For the following system of linear equations (in w, x, y and z):
(a) write as an augmented matrix; (b) convert to reduced echelon form; (c) find all solutions.
w + x + y + z = 5
w + 2x + y + 2z = 10
-w + x + y – z = -3
3w + 4x – 5y – 5z = 1 [10 marks]
2. Using your answer from question 1, find all solutions to the following linear system.
w + x + y + z + v = 5
w + 2x + y + 2z + 2v = 10
-w + x + y – z – v = -3
3w + 4x – 5y – 5z – 5v = 1 [4 marks]
3. Determine the values of k so that the following system of linear equations (in x, y and z) has:
(a) a unique solution; (b) no solution; (c) an infinite number of solutions.
2x + (k + 1)y + 2z = 3
2x + 3y + kz = 3
3x + 3y – 3z = 3 [10 marks]
4. Suppose that O, A and B are three non-collinear points in a plane.
Let OP = 2OB – OA, OQ = 2OA – 3OB, OR = 3OB – OA and OS = OA + OB.
(a) Write a vector equation for the line that passes through Q and is parallel to the vector
OP, in terms of the vectors OA and OB.
(b) Determine whether or not the point R lies on the line from part (a).
(c) Determine whether or not the point S lies on the line from part (a).
(d) Show algebraically that the vectors OS and OP –
3
4
OR are linearly dependent.
(e) Find the point of intersection of the line through O and B and the line from part (a).
(f) Show that the line through O and P does not intersect the line from part (a).
[16 marks]
5. (a) Find a parametric vector equation (in terms of coordinates) that represents the line in
3-space described by the Cartesian equations x – 3z = -8 and -2x + 4y – 2z = 0.
(b) Find the intersection of the line through the point P(3, 0, -1), that is parallel to the vector
OA, whose coordinates are (1, -1, -2), with the line through the points Q(3, 4, 5) and
R(-3, -2, -1).
(c) Then find a Cartesian equation that describes the plane that contains these two lines.

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