Chapter 7 Interactive Lab

Solving Backwards

A matrix sends inputs to outputs. A linear system asks the reverse question: which input produced the output?

1. Matrix machine: forward and backward

Choose the hidden input vector. The matrix sends it forward to an output. Then solve backwards to recover the input.

A = [[2, 1], [1, 3]],   b = A x
Output b₁ = 2x₁ + x₂
Output b₂ = x₁ + 3x₂
Recovered input

The blue arrow is the input x. The purple arrow is the output b.

2. Equations as lines

The system Ax = b can be drawn as two lines. The solution is their intersection.

System:

2x₁ + x₂ = b₁
x₁ + 3x₂ = b₂
Intersection / solution

3. Column picture: the solution is a recipe

Solving Ax = b means building b from the columns of A. The entries of x are recipe coefficients.

x₁ [2,1] + x₂ [1,3] = b
Built target b

The first arrow follows x₁ times column 1. The second arrow follows x₂ times column 2. Together they build b.

4. The three possible outcomes

A system can have exactly one solution, no solution, or infinitely many solutions.

5. Elimination step by step

Row operations simplify a system without changing its solution set.

6. Sensitivity: when solving backwards is fragile

Some matrix machines can be reversed, but only with difficulty. Small changes in the output can cause large changes in the recovered input.

A = [[1, 1], [1, 1 + ε]]
Recovered x
Amplification

7. Reflection and design challenge

Write your own reverse-engineering story. What are the hidden quantities? What observations are visible? What matrix connects them?

A strong answer should include A, x, b, and an interpretation of the recovered x.