Step by step
The question
Question: work out (1, 0, 2) + (3, 4, −1).
[1; 0; 2] + [3; 4; −1]
A vector in R 3 is a column of 3 numbers.
Numbers times vectors, added together, is called a linear combination.
a vector in R³: a column of 3 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.
= [1 + 3; 0 + 4; 2 + (−1)]
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 (4, 4, 1).
= [4; 4; 1]
2 Check
Let's check.
We take the second part away from the answer.
The first part must come back.
= [4; 4; 1] − [3; 4; −1]
= [4 − 3; 4 − 4; 1 − (−1)]
= [1; 0; 2]
We get (1, 0, 2): that is (1, 0, 2), the first part.
So the answer is right!
Let's look back.
First: add matching entries.
Last: check.
Answer
= [4; 4; 1]