Calculate 2(1, -1, 0, 3) - (3, 0, 2, 1) Step by Step

Step by step

Question: work out 2(1, −1, 0, 3) − (3, 0, 2, 1).

2 × [1; −1; 0; 3] − [3; 0; 2; 1]

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

Each vector is multiplied by its number: every entry.

2 × 1 at the top, and so on down.

= [2 × 1; 2 × (−1); 2 × 0; 2 × 3] − [3; 0; 2; 1]

We work out each times.

= [2; −2; 0; 6] − [3; 0; 2; 1]

Now matching entries go together: top with top, then the next, down to the bottom.

A minus between the vectors takes away every entry.

= [2 − 3; −2 − 0; 0 − 2; 6 − 1]

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, −2, −2, 5).

= [−1; −2; −2; 5]

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.

= [−1; −2; −2; 5] + [3; 0; 2; 1]

= [−1 + 3; −2 + 0; −2 + 2; 5 + 1]

= [2; −2; 0; 6]

We get (2, −2, 0, 6): that is 2 × (1, −1, 0, 3), 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

= [−1; −2; −2; 5]