Step by step
The question
Question: work out (4, 7) − (1, 2).
[4; 7] − [1; 2]
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.
A minus between the vectors takes away every entry.
= [4 − 1; 7 − 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
So the answer is the vector (3, 5).
= [3; 5]
2 Check
Let's check.
We take the second part away from the answer (taking away a minus part is adding it back).
The first part must come back.
= [3; 5] + [1; 2]
= [3 + 1; 5 + 2]
= [4; 7]
We get (4, 7): that is (4, 7), the first part.
So the answer is right!
Let's look back.
First: add matching entries.
Last: check.
Answer
= [3; 5]