Step by step
The question
Question: work out 3(2, −1) + 2(1, 4).
3 × [2; −1] + 2 × [1; 4]
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.
3 × 2 at the top, and so on down.
= [3 × 2; 3 × (−1)] + [2 × 1; 2 × 4]
We work out each times.
= [6; −3] + [2; 8]
2 Add matching entries
Now matching entries go together: top with top, then the next, down to the bottom.
= [6 + 2; −3 + 8]
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 (8, 5).
= [8; 5]
3 Check
Let's check.
We take the second part away from the answer.
The first part must come back.
= [8; 5] − [2; 8]
= [8 − 2; 5 − 8]
= [6; −3]
We get (6, −3): that is 3 × (2, −1), 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
= [8; 5]