Step by step
The question
Question: find where A sends (4, −1).
A = [1, 2; 0, 1], v = [4; −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.
= [1, 2; 0, 1] × [4; −1]
= [1 × 4 + 2 × (−1); 0 × 4 + 1 × (−1)]
We work out every times.
= [4 − 2; 0 − 1]
= [2; −1]
So the image is (2, −1).
A v = [2; −1]
2 Draw it
v is the arrow to (4, −1).
A v is the arrow to (2, −1).
A is a shear across by 2, 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 4 times column 1, plus (−1) times column 2.
= 4 × [1; 0] − [2; 1]
= [4 × 1; 4 × 0] − [2; 1]
= [4; 0] − [2; 1]
= [4 − 2; 0 − 1]
= [2; −1]
We get (2, −1) again.
So the answer is right!
Let's look back.
First: multiply.
Then: draw it.
Last: check.
Answer
A v = [2; −1]