Practice Homogeneous Systems (25.5.1) - Solutions of Linear Systems: Existence, Uniqueness, General Form

Practice: Homogeneous Systems

Lesson 25.5.1Solutions of Linear Systems: Existence, Uniqueness, General Form

Practise this section

9 questions

Flashcard drill
3 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
Sign up to start the exercise
Quiz
3 questions · about 2 min · pass mark 70%. Hints available.
Sign up to start the quiz

6 sample questions

Question 1.
What defines a homogeneous system of equations?
Show hintHide hint
Consider the form of the system.
Question 2.
What is the trivial solution to a homogeneous system?
Show hintHide hint
Think about the simplest solution.
3Question 3.
What is a homogeneous system of linear equations?
- A system where Ax = 0
- A system where b ≠ 0
- A system with unique solutions
Show hintHide hint
Recall the definition we discussed.
4Question 4.
True or False: The trivial solution of a homogeneous system is x = 1.
- True
- False
Show hintHide hint
Think about the simplest solution again.
Question 5.
Given a 3x3 matrix A with rows that are linearly dependent, find the null space of A and discuss the implications on the solutions of Ax = 0.
Show hintHide hint
Look for combinations of rows that demonstrate linear dependency.
Question 6.
If a homogeneous system Ax = 0 has three variables and the rank of A is 1, describe the geometric interpretation of the solution space.
Show hintHide hint
Draw the axes and think about which direction represents the span of the solution.

5 more questions in this section. Enroll free to practise them all.