What does [1, 0; 3, 1] do to the unit square Step by Step

Step by step

The question

Question: find what A does to the unit square.

A = [1, 0; 3, 1]

Times A turns every point into a new point.

So A moves the whole plane.

The unit square has corners (0, 0), (1, 0), (1, 1) and (0, 1).

A × (point) = new point:

A moves every point of the plane

1 Where (1, 0) goes

First, where does (1, 0) go?

A times (1, 0).

= [1, 0; 3, 1] × [1; 0]

= [1 × 1 + 0 × 0; 3 × 1 + 1 × 0]

= [1; 3]

(1, 3): that is the first column of A.

2 Where (0, 1) goes

Now (0, 1).

A times (0, 1).

= [1, 0; 3, 1] × [0; 1]

= [1 × 0 + 0 × 1; 3 × 0 + 1 × 1]

= [0; 1]

(0, 1): the second column of A.

Careful!

The columns of A show where (1, 0) and (0, 1) go.

Not the rows.

Careful: columns, not rows

3 Draw the new shape

The unit square, dashed.

(1, 0) goes to (1, 3).

(0, 1) goes to (0, 1).

The square goes to the shape with these two sides.

Its far corner is (1, 4).

4 Name it

(0, 1) stays.

(1, 0) slides up to (1, 3).

The square leans up: a shear.

Its width stays 1.

So A is a shear up by 3.

= a shear up by 3

The determinant tells the area: det A = 1.

The square's area 1 becomes 1.

det A = 1 × 1 − 3 × 0 = 1

area: 1 → 1

5 Check

Let's check the far corner: A times (1, 1).

= [1, 0; 3, 1] × [1; 1]

= [1 × 1 + 0 × 1; 3 × 1 + 1 × 1]

= [1; 4]

We get (1, 4): the far corner of the new shape, column 1 plus column 2.

So the picture is right!

Let's look back.

First: where (1, 0) goes.

Then: where (0, 1) goes.

Then: draw the new shape.

Then: name it.

Last: check.

Answer