Page 321 - 35Linear Algebra

D

Sample First Midterm

Here are some worked problems typical for what you might expect on a first
midterm examination.

1. Solve the following linear system. Write the solution set in vector form.
Check your solution. Write one particular solution and one homogeneous
solution, if they exist. What does the solution set look like geometrically?

x + 3y = 4
x − 2y + z = 1
2x + y + z = 5

2. Consider the system of equations

x − z + 2w = −1




 x + y + z − w = 2


− y − 2z + 3w = −3




5x + 2y −

 z + 4w = 1
(a) Write an augmented matrix for this system.
(b) Use elementary row operations to find its reduced row echelon form.
(c) Write the solution set for the system in the form
X
S = {X 0 + µ i Y i : µ i ∈ R}.
i

321