Step by step
The question
Question: find where A sends (3, 1).
A = [0, −1; 1, 0], v = [3; 1]
The image of v is where A sends it: A times v.
image of v: the new point A × v
1 Multiply
We work out A times v.
= [0, −1; 1, 0] × [3; 1]
= [0 × 3 + (−1) × 1; 1 × 3 + 0 × 1]
We work out every times.
= [0 − 1; 3 + 0]
= [−1; 3]
So the image is (−1, 3).
A v = [−1; 3]
2 Draw it
v is the arrow to (3, 1).
A v is the arrow to (−1, 3).
A is a turn of 90° counterclockwise, and the arrow moved that way.
Careful!
Each entry of A v is a row of A times v: first times first, second times second.
Careful: row times column, in order
3 Check
Let's check another way: A v is 3 times column 1, plus 1 times column 2.
= 3 × [0; 1] + [−1; 0]
= [3 × 0; 3 × 1] + [−1; 0]
= [0; 3] + [−1; 0]
= [0 + (−1); 3 + 0]
= [−1; 3]
We get (−1, 3) again.
So the answer is right!
Let's look back.
First: multiply.
Then: draw it.
Last: check.
Answer
A v = [−1; 3]