Calculate (4, 2) - 2(2, 1) Step by Step

Step by step

Question: work out (4, 2) − 2(2, 1).

[4; 2] − 2 × [2; 1]

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

Each vector is multiplied by its number: every entry.

2 × 2 at the top, and so on down.

= [4; 2] − [2 × 2; 2 × 1]

We work out each times.

= [4; 2] − [4; 2]

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 − 4; 2 − 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 (0, 0).

= [0; 0]

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.

= [0; 0] + [4; 2]

= [0 + 4; 0 + 2]

= [4; 2]

We get (4, 2): that is (4, 2), 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

= [0; 0]