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