Step by step
The question
Question: work out (1, 2, 3, 4) + (0, −1, 2, 5).
[1; 2; 3; 4] + [0; −1; 2; 5]
A vector in R 4 is a column of 4 numbers.
Numbers times vectors, added together, is called a linear combination.
a vector in R⁴: a column of 4 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 + 0; 2 + (−1); 3 + 2; 4 + 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 (1, 1, 5, 9).
= [1; 1; 5; 9]
2 Check
Let's check.
We take the second part away from the answer.
The first part must come back.
= [1; 1; 5; 9] − [0; −1; 2; 5]
= [1 − 0; 1 − (−1); 5 − 2; 9 − 5]
= [1; 2; 3; 4]
We get (1, 2, 3, 4): that is (1, 2, 3, 4), the first part.
So the answer is right!
Let's look back.
First: add matching entries.
Last: check.
Answer
= [1; 1; 5; 9]