Step by step
The question
Question: find where A sends (2, 3).
A = [1, 0; 0, −1], v = [2; 3]
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.
= [1, 0; 0, −1] × [2; 3]
= [1 × 2 + 0 × 3; 0 × 2 + (−1) × 3]
We work out every times.
= [2 + 0; 0 − 3]
= [2; −3]
So the image is (2, −3).
A v = [2; −3]
2 Draw it
v is the arrow to (2, 3).
A v is the arrow to (2, −3).
A is a flip in the x-axis, 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 2 times column 1, plus 3 times column 2.
= 2 × [1; 0] + 3 × [0; −1]
= [2 × 1; 2 × 0] + [3 × 0; 3 × (−1)]
= [2; 0] + [0; −3]
= [2 + 0; 0 + (−3)]
= [2; −3]
We get (2, −3) again.
So the answer is right!
Let's look back.
First: multiply.
Then: draw it.
Last: check.
Answer
A v = [2; −3]