Step by step
The question
Question: find where A sends (−2, 3).
A = [3, 0; 0, 2], 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.
= [3, 0; 0, 2] × [−2; 3]
= [3 × (−2) + 0 × 3; 0 × (−2) + 2 × 3]
We work out every times.
= [−6 + 0; 0 + 6]
= [−6; 6]
So the image is (−6, 6).
A v = [−6; 6]
2 Draw it
v is the arrow to (−2, 3).
A v is the arrow to (−6, 6).
A is a stretch: across by 3, up 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 −2 times column 1, plus 3 times column 2.
= −2 × [3; 0] + 3 × [0; 2]
= −[2 × 3; 2 × 0] + [3 × 0; 3 × 2]
= −[6; 0] + [0; 6]
= [−6 + 0; −0 + 6]
= [−6; 6]
We get (−6, 6) again.
So the answer is right!
Let's look back.
First: multiply.
Then: draw it.
Last: check.
Answer
A v = [−6; 6]