Checkpoint: matrices, bases, and eigenvector directions

Where we are on the map

This closes the linear algebra block; a column of numbers is insufficient without dimensions, geometric meaning, and a check against the original transformation.

Begin with a familiar image

Camera settings, a sound mixer, and a network model can share the same matrix form. The mathematics is reliable when rows, columns, coefficients, and basis have clear meanings.

Precise meaning

Mastery means seeing one object as a coefficient table, an action on a vector, a geometric transformation, and a set of checkable equations.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

dimensions → action → space and basis → special directions → meaning → direct check

Worked example

Matrix A=[[2,1],[1,2]] transforms vectors, while the basis (1,1), (1,−1) reveals the action as independent scale factors 3 and 1.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Trusting λ and v without verification is wrong. The equality Av=λv must hold component by component.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Complete ten problems and show calculation and verification.

1. For a 3×2 matrix A and a 2×4 matrix B, find the size of AB and explain why BA does not exist.
2. Find Ax for A=[[1,2],[3,−1]], x=(2,4).
3. Verify A(u+v)=Au+Av for A=[[2,0],[1,3]], u=(1,−1), v=(4,2).
4. Solve 2x+y=7, x−y=2 and check by substitution.
5. Are (1,2) and (3,6) independent? Describe their span.
6. Decompose (7,1) in basis b₁=(1,1), b₂=(2,−1).
7. Show that (1,0), (0,1), (1,1) are dependent.
8. Name eigenvalues and directions of D=[[5,0],[0,−1]].
9. Verify both eigenpairs of A=[[2,1],[1,2]].
10. Build a real two-input, two-output matrix model and state units and one limitation.

Pass target: at least 8/10, including problem 6 or 10. After seven days, complete a changed version with a target of at least 7/10.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The next block counts possibilities, measures uncertainty, and draws cautious conclusions from data.

Course contents