10 questions covering the entire module. Work each problem on paper, then reveal the answer to check yourself.
1
Which of the following is NOT a linear equation? (a) 3x - 2y = 5 (b) x^2 + y = 3 (c) x + y + z = 0
(b) x^2 + y = 3 is not linear because x is squared.
2
Write the augmented matrix for: x - 3y = 2 and 4x + y = 9.
[ 1 -3 | 2 ] / [ 4 1 | 9 ]
3
Name the three elementary row operations.
(1) Row swap (interchange two rows), (2) Row scaling (multiply a row by a nonzero constant), (3) Row replacement (add a multiple of one row to another).
4
After performing R_2 - 4R_1 on [ 1 -3 | 2 ] / [ 4 1 | 9 ], what is the new row 2?
The RREF of a system is [ 1 0 0 | 2 ] / [ 0 1 0 | -1 ] / [ 0 0 0 | 3 ]. What is the solution?
No solution. Row 3 says 0 = 3, which is a contradiction. The system is inconsistent.
8
A system has 4 unknowns and its RREF has 2 pivot columns. How many free variables are there? What type of solution set is it (assuming consistency)?
4 - 2 = 2 free variables. Infinitely many solutions, forming a plane (2-dimensional family) in R^4.
9
Given RREF [ 1 0 2 | 4 ] / [ 0 1 -1 | 3 ], write the solution in parametric vector form.
Free: x_3 = t. x_1 = 4 - 2t, x_2 = 3 + t. x = (4, 3, 0) + t(-2, 1, 1).
10
True or False: A homogeneous system with more unknowns than equations always has a nontrivial solution.
True. With more unknowns than equations, there are at most as many pivots as equations, so at least one free variable exists, guaranteeing infinitely many (nontrivial) solutions.