Matrices: a table of numbers that transforms data

Where we are on the map

A two-by-two matrix transforms vector (4, 2) into vector (10, 10)

Begin with a familiar image

A control panel with two inputs and two displays mixes signals: each display combines the inputs with its own coefficients. Its settings table is a matrix.

Precise meaning

A matrix is a rectangular table of numbers. An m×n matrix has m rows and n columns. In A·x, every row of A forms a dot product with column x. The operation exists only when the number of columns of A equals the number of components of x.

The lesson’s support signal

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

dimensions → row · column → one component → every row → new vector → meaning check

Worked example

Let A=[[2,1],[1,3]] and x=(4,2). The first row gives 2·4+1·2=10; the second gives 1·4+3·2=10. Thus Ax=(10,10). Dimensions confirm (2×2)·(2×1)=(2×1).

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

Multiplying a 2×3 matrix by a two-component vector is invalid. The inner dimensions 3 and 2 do not match.

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. For A=[[3,−1],[2,4]], u=(5,2), and v=(−1,3), find Au, Av, and A(u+v). Verify A(u+v)=Au+Av and state every dimension.

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 lesson uses a basis to explain which vectors can be built from selected directions.

Course contents