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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Parameterize a nullspace and connect free variables to lost information.
- Justify the conclusion "Null(A)=span{(−1,−1,1)}; nullity=1" using the stated assumptions.
Before you start
Linear systems and vectors.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find the nullspace of A=[[1,0,1],[0,1,1]].
Why this math matters
Parameterize a nullspace and connect free variables to lost information. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The map has real inputs and outputs.
- The two rows are independent.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find all invisible inputs of a linear map
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the nullspace of A=[[1,0,1],[0,1,1]].
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Ax=0 gives x₁+x₃=0 and x₂+x₃=0
Each row contributes an independent constraint.
Work through the mathematics
Let x₃=t; then x=(−t,−t,t)
The third variable is free, so every solution is a multiple of one vector.
Check the conclusion
Null(A)=span{(−1,−1,1)}; nullity=1
Multiplying this vector by A gives zero; two pivots plus one free direction account for three input coordinates.
The result
Null(A)=span{(−1,−1,1)}; nullity=1
Multiplying this vector by A gives zero; two pivots plus one free direction account for three input coordinates.
Common mistakes to catch
- A nullspace contains vectors, not just a count.
- The zero vector alone cannot serve as a basis vector.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find all x with Ax=(2,3).
Show a hint
Add the nullspace to one particular solution.
Reveal answer and explanation
x=(2,3,0)+t(−1,−1,1)
Adding an invisible input preserves the measured output.
Practice 2
Is the nullspace of an invertible square matrix nontrivial?
Show a hint
Apply its inverse to Ax=0.
Reveal answer and explanation
No; it contains only zero
Multiplication by A⁻¹ gives x=0.
Take the idea with you
Interpret nullspace directions as changes a sensor system cannot detect.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Describe the same vector in another coordinate system
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