What does [1, -1; 1, 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, −1; 1, 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, −1; 1, 1] × [1; 0]

= [1 × 1 + (−1) × 0; 1 × 1 + 1 × 0]

= [1; 1]

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

2 Where (0, 1) goes

Now (0, 1).

A times (0, 1).

= [1, −1; 1, 1] × [0; 1]

= [1 × 0 + (−1) × 1; 1 × 0 + 1 × 1]

= [−1; 1]

(−1, 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, 1).

(0, 1) goes to (−1, 1).

The square goes to the shape with these two sides.

Its far corner is (0, 2).

4 Name it

It is not one simple move.

The square becomes a slanted shape with corners (0, 0), (1, 1), (0, 2) and (−1, 1).

= a mix: the square becomes a slanted shape

The determinant tells the area: det A = 2.

The square's area 1 becomes 2.

det A = 1 × 1 − 1 × (−1) = 2

area: 1 → 2

5 Check

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

= [1, −1; 1, 1] × [1; 1]

= [1 × 1 + (−1) × 1; 1 × 1 + 1 × 1]

= [0; 2]

We get (0, 2): 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