Step by step
The question
Question: work out 3(1, 2, 0).
3 × [1; 2; 0]
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 Multiply each vector by its number
Each vector is multiplied by its number: every entry.
3 × 1 at the top, and so on down.
= [3 × 1; 3 × 2; 3 × 0]
We work out each times.
= [3; 6; 0]
2 Add matching entries
So the answer is the vector (3, 6, 0).
= [3; 6; 0]
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.
= [3; 6; 0]
= [3; 6; 0]
We get (3, 6, 0): that is 3 × (1, 2, 0), 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