Calculate 2a - 3b for a = (1, -2, 0, 4), b = (2, -1, 1, 3)

Step by step

The question

Question: work out 2a − 3b.

a is (1, −2, 0, 4), and b is (2, −1, 1, 3).

We put them in.

So 2a − 3b is this.

A vector in R 4 is a column of 4 numbers.

Numbers times vectors, added together, is called a linear combination.

1 Multiply each vector by its number

Each vector is multiplied by its number: every entry.

2 × 1 at the top, and so on down.

We work out each times.

2 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.

Careful!

Taking away a negative is adding: −4 − (−3) = −4 + 3.

So the answer is the vector (−4, −1, −3, −1).

3 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.

We get (2, −4, 0, 8): that is 2 × a, 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.