COSMO · STUDY + TEACHING DECKLinear Algebra
November 27, 2025
C. A question worth sharingVectors and Matrices
Use coordinates and matrices to represent directions and transformations.
ThinkCompareExplain
Today’s routeA question becomes an explanation.
Try
Enter through a problem. A partial attempt counts.
Connect
Use vector, matrix, linear transformation to name what you notice.
Convince
Build an argument a partner can check.
The goal is a method someone else can use.
The openerStart with a question.
Opening problem
Add vectors (3,-1) and (-2,5).
Think · 2 min
Sketch, test a small case, or make a first guess.
Pair · 3 min
Compare methods. Find one thing you agree on and one question.
Be ready to share your route, even if it is unfinished.
Compare before checkingPut two attempts in conversation.
Explain
Tell a partner how you started. Let them finish before responding.
Question
Which step feels certain? Which step still needs evidence?
Build
Combine one useful idea from each attempt into a new route.
Can you explain your partner’s method in your own words?
Board conversationNotice before naming.
Look for structure
- 01Try a small case and record every step.
- 02Separate what changes from what stays fixed.
- 03State the pattern in a sentence before naming the theorem.
Make a conjecture
We think ___ because ___. A case that might break this is ___.
What changed? What stayed the same?
Useful languageNow the language has a job.
vector
quantity with components
matrix
rectangular array defining a linear map
linear transformation
map preserving addition and scaling
Use one of these words to explain something you already noticed.
The core ideaKeep the idea. Explain the conditions.
A(Bv)=(AB)v The idea
Use coordinates and matrices to represent directions and transformations.
What to remember
A matrix is a transformation recipe, not merely a table of numbers.
What information does this method keep?
Worked exampleOne route, one decision at a time
The problem
Apply matrix [[2,0],[0,-1]] to vector (3,4).
Predict each next step
- 01Multiply the first row by the vector to get 6.
- 02Multiply the second row to get -4.
- 03The image is (6,-4).
- 04This stretches x and reflects y.
Which step needs a reason, not just a calculation?
See the structureThe structure behind vectors and matrices
Read the representation
Use coordinates and matrices to represent directions and transformations.
Look together
Point to the part that explains the relationship. Change one input and predict what changes.
What can you see here that was harder to see in words?
Two approachesCompare the reasoning.
Structural method
- 01Name the objects and conditions.
- 02Use a reusable relationship.
- 03Explain why the method must work.
Brute force
- 01May help with a small test case.
- 02Can hide the general reason.
- 03Becomes unreliable when the scale grows.
Which route can you justify? What would you test first?
Reasoning clinicFind the step that needs repair.
Investigate these concerns
- 01May help with a small test case.
- 02Can hide the general reason.
- 03Becomes unreliable when the scale grows.
Repair together
Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.
A useful correction explains why the original step fails.
Work togetherInvestigate vectors and matrices
Your shared task
Groups transform a coordinate shape and infer the matrix from where basis vectors move.
Explore
4 min - test cases
Compare
5 min - form a rule
Defend
4 min - defend the rule
Everyone should be able to explain the group’s final claim.
Share → connectWhat did the room discover?
Bring your evidence
Share a conjecture, a useful example, or a route that changed.
Connections to investigate
- 01A matrix is a transformation recipe, not merely a table of numbers.
- 02A complete solution names the condition that makes the method valid.
- 03A counterexample is useful when it exposes a missing condition.
Which observation explains the mechanism?
Try → explain → checkYour turn to build the route
The problem
Find the determinant of [[3,2],[1,4]].
Predict each next step
- 01Compute ad-bc.
- 023(4)-2(1)=12-2.
- 03The determinant is 10.
- 04Area scales by a factor of 10.
Which step needs a reason, not just a calculation?
Convince a skepticTurn a pattern into a proof.
The claim
Matrix multiplication represents composition of linear transformations.
Build the argument
- 01Apply the second transformation to a vector first.
- 02Apply the first transformation to that result.
- 03The combined coefficients are exactly the entries of the product matrix.
What does each step rely on? Where are the conditions used?
Stretch togetherTake the idea somewhere new.
Transfer challenge
Find a matrix that rotates the plane 90 degrees counterclockwise.
Choose a route
Name the objects, the conditions, and a relationship that might help. Start with one test case.
What transfers from the earlier example? What is different?
Peer reviewCheck the argument together.
Restate
Explain the other person’s route before evaluating it.
Test
Try a small case or boundary case. Does every step still make sense?
Revise
Fix an assumption, add a reason, or find a second method.
Where did your thinking change?
Beyond this problemWhere might this idea help?
A connection
Vectors and matrices support graphics, data science, physics, optimization, and networks.
Discuss the model
- 01What objects or quantities are represented?
- 02Which relationship from today is preserved?
- 03What assumption would you check before trusting it?
Invent one more situation with the same structure.
Exit ticketLeave with an explanation.
Question 1
Add vectors (3,-1) and (-2,5).
Question 2
Explain why this claim is true: Matrix multiplication represents composition of linear transformations.
Question 3
Find a matrix that rotates the plane 90 degrees counterclockwise.
Choose one. Write an answer with a reason, then compare with a partner.
Reflect togetherMake your thinking visible.
I used to think…
Name a claim or approach that changed.
Now I can explain…
Give the reason, not only the result.
I still wonder…
Write a question worth bringing to the next circle.
Share one sentence. Ask one follow-up question.
The idea to keepA matrix is a transformation recipe, not merely a table of numbers.
Use the matching practice sets to move from direct examples to proof and transfer.
A good question belongs to everyone in the room. 1 / 21
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