Step by step
The question
Question: work out −2(1, 3) + (5, 2).
−2 × [1; 3] + [5; 2]
A vector in R 2 is a column of 2 numbers.
Numbers times vectors, added together, is called a linear combination.
a vector in R²: a column of 2 numbers
linear combination: numbers × vectors, added
1 Multiply each vector by its number
Each vector is multiplied by its number: every entry.
2 × 1 at the top, and so on down.
= −[2 × 1; 2 × 3] + [5; 2]
We work out each times.
= −[2; 6] + [5; 2]
2 Add matching entries
Now matching entries go together: top with top, then the next, down to the bottom.
= [−2 + 5; −6 + 2]
Careful!
Every entry gets the same treatment.
Top with top, second with second, down to the bottom.
Careful: every entry: top with top, down to the bottom
So the answer is the vector (3, −4).
= [3; −4]
3 Check
Let's check.
We take the second part away from the answer.
The first part must come back.
= [3; −4] − [5; 2]
= [3 − 5; −4 − 2]
= [−2; −6]
We get (−2, −6): that is −2 × (1, 3), the first part.
So the answer is right!
Let's look back.
First: multiply each vector by its number.
Then: add matching entries.
Last: check.
Answer
= [3; −4]