Step by step
The question
Question: work out (3, 1) + (2, 5).
[3; 1] + [2; 5]
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 Add matching entries
Now matching entries go together: top with top, then the next, down to the bottom.
= [3 + 2; 1 + 5]
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 (5, 6).
= [5; 6]
2 Check
Let's check.
We take the second part away from the answer.
The first part must come back.
= [5; 6] − [2; 5]
= [5 − 2; 6 − 5]
= [3; 1]
We get (3, 1): that is (3, 1), the first part.
So the answer is right!
Let's look back.
First: add matching entries.
Last: check.
Answer
= [5; 6]