Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Separate elimination from back substitution to solve repeated linear systems.
- Justify the conclusion "Ux=y gives x=(2,1); Ax=(5,11)" using the stated assumptions.
Before you start
Matrix multiplication and two-variable equations.
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
Factor A=[[2,1],[4,3]] and solve Ax=(5,11).
Why this math matters
Separate elimination from back substitution to solve repeated linear systems. 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:
- All arithmetic is exact real arithmetic.
- The first pivot is nonzero in this example.
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.
Reuse elimination with an LU factorization
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Factor A=[[2,1],[4,3]] and solve Ax=(5,11).
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
R₂←R₂−2R₁ gives U=[[2,1],[0,1]]
The elimination multiplier is two; storing it avoids repeating elimination for another right-hand side.
Work through the mathematics
L=[[1,0],[2,1]]; Ly=(5,11) gives y=(5,1)
Forward substitution subtracts twice the first component from the second.
Check the conclusion
Ux=y gives x=(2,1); Ax=(5,11)
Back substitution gives x₂=1 and x₁=2, and direct multiplication checks the factorization-based solution.
The result
Ux=y gives x=(2,1); Ax=(5,11)
Back substitution gives x₂=1 and x₁=2, and direct multiplication checks the factorization-based solution.
Common mistakes to catch
- The multiplier goes in L, not its negative.
- LU is matrix multiplication, not entrywise multiplication.
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
Use the same factors for b=(3,7).
Show a hint
First solve Ly=b.
Reveal answer and explanation
x=(1,1)
y=(3,1), then 2x₁+x₂=3 and x₂=1.
Practice 2
What if the first pivot is zero?
Show a hint
Can division by that pivot work?
Reveal answer and explanation
Row pivoting may be required
LU without permutations is not available for every invertible matrix.
Take the idea with you
Explain why one factorization helps when a structure is tested under many load vectors.
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: Find all invisible inputs of a linear map
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