Step by step
The question
Question: find where A sends (1, 2).
A = [2, 1; 1, 1], v = [1; 2]
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.
= [2, 1; 1, 1] × [1; 2]
= [2 × 1 + 1 × 2; 1 × 1 + 1 × 2]
We work out every times.
= [2 + 2; 1 + 2]
= [4; 3]
So the image is (4, 3).
A v = [4; 3]
2 Draw it
v is the arrow to (1, 2).
A v is the arrow to (4, 3).
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 1 times column 1, plus 2 times column 2.
= [2; 1] + 2 × [1; 1]
= [2; 1] + [2 × 1; 2 × 1]
= [2; 1] + [2; 2]
= [2 + 2; 1 + 2]
= [4; 3]
We get (4, 3) again.
So the answer is right!
Let's look back.
First: multiply.
Then: draw it.
Last: check.
Answer
A v = [4; 3]