Lab 1. Linear Systems: Interactive Practice
This lab accompanies Chapter 1: Linear Systems.
The goal is to connect three views of a linear system:
- Algebra: the equations and row-reduction.
- Geometry: the intersection of lines.
- Computation: rank tests and numerical/symbolic checking.
Python practice notebook
You may also use the Jupyter notebook version for longer Python practice:
Interactive lab
Submission questions
Submit short written answers to the following questions.
Question 1. Unique solution
Use the interactive lab to create a system with exactly one solution.
Record:
- the two equations;
- the coefficient matrix \(A\);
- the vector \(b\);
- the augmented matrix \([A \mid b]\);
- \(\operatorname{rank}(A)\);
- \(\operatorname{rank}([A \mid b])\);
- the solution \((x,y)\).
Explain why the system has a unique solution both algebraically and geometrically.
Question 2. No solution
Use the interactive lab to create a system with no solution.
Record the same information as in Question 1.
Explain why
\[ \operatorname{rank}(A) < \operatorname{rank}([A \mid b]). \]
Also explain the geometry of the two lines.
Question 3. Infinitely many solutions
Use the interactive lab to create a system with infinitely many solutions.
Record the same information as in Question 1.
Explain why
\[ \operatorname{rank}(A)=\operatorname{rank}([A \mid b])<2. \]
Also explain why the two equations describe the same line.
Question 4. Change only the right-hand side
Start from a system with infinitely many solutions. Then change only \(c_2\).
What changes geometrically? What changes algebraically? What happens to the ranks?
Question 5. AI companion
Ask an AI tool:
Explain the rank test for consistency of a linear system \(Ax=b\).
Then critique the answer.
Your critique should answer:
- Is the statement mathematically correct?
- Does it distinguish \(A\) from \([A \mid b]\)?
- Does it explain the geometric meaning?
- Can you give a concrete example where the system has no solution?