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Linear combinations

Linear Algebra · 21 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Multiply each vector by its number
  2. Add matching entries
  3. Check

The steps for Calculate 2(1, 0, 3) - 3(2, 1, -1). Every solution names its steps, so you always know what comes next.

Worked examples

Calculate (3, 1) + (2, 5) Easy

The question

Question: work out (3, 1) + (2, 5).

[3; 1] + [2; 5]

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.

= [3 + 2; 1 + 5]

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 (5, 6).

= [5; 6]

2 Check

Let's check.

We take the second part away from the answer.

The first part must come back.

= [5; 6] − [2; 5]
= [5 − 2; 6 − 5]
= [3; 1]

We get (3, 1): that is (3, 1), the first part.

So the answer is right!

Let's look back.

First: add matching entries.

Last: check.

Answer
= [5; 6]

▶ Watch it solved

Calculate 2(1, 0, 3) - 3(2, 1, -1) Medium

The question

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

2 × [1; 0; 3] − 3 × [2; 1; −1]

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.

2 × 1 at the top, and so on down.

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

We work out each times.

= [2; 0; 6] − [6; 3; −3]

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.

= [2 − 6; 0 − 3; 6 − (−3)]

Careful!

Taking away a negative is adding: 6 − (−3) = 6 + 3.

Careful: not 6 − 3 ✗

So the answer is the vector (−4, −3, 9).

= [−4; −3; 9]

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.

= [−4; −3; 9] + [6; 3; −3]
= [−4 + 6; −3 + 3; 9 + (−3)]
= [2; 0; 6]

We get (2, 0, 6): that is 2 × (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
= [−4; −3; 9]

▶ Watch it solved

Calculate u + 2v - w for u = (1, 0, 2), v = (0, 1, -1), w = (3, 1, 1) Harder

The question

Question: work out u + 2v − w.

u = [1; 0; 2]
v = [0; 1; −1]
w = [3; 1; 1]

u is (1, 0, 2), and v is (0, 1, −1), and w is (3, 1, 1).

We put them in.

So u + 2v − w is this.

= [1; 0; 2] + 2 × [0; 1; −1] − [3; 1; 1]

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.

2 × 0 at the top, and so on down.

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

We work out each times.

= [1; 0; 2] + [0; 2; −2] − [3; 1; 1]

2 Add matching entries

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

= [1 + 0 − 3; 0 + 2 − 1; 2 + (−2) − 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 (−2, 1, −1).

= [−2; 1; −1]

3 Check

Let's check.

We take the other parts away from the answer (taking away a minus part is adding it back).

The first part must come back.

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

We get (1, 0, 2): that is u, 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
= [−2; 1; −1]

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. (3, 1) + (2, 5)
    Answer
    = [5; 6]
  2. (4, 7) - (1, 2)
    Answer
    = [3; 5]
  3. 2(3, -1)
    Answer
    Careful: every entry: top with top, down to the bottom
  4. (1, 0, 2) + (3, 4, -1)
    Answer
    = [4; 4; 1]
  5. 3(1, 2, 0)
    Answer
    Careful: every entry: top with top, down to the bottom
  6. (5, 2, 3) - (2, 2, 1)
    Answer
    = [3; 0; 2]
  7. -2(3, 1)
    Answer
    Careful: every entry: top with top, down to the bottom

Medium

  1. 2(1, 0, 3) - 3(2, 1, -1)
    Answer
    = [−4; −3; 9]
  2. 3(2, -1) + 2(1, 4)
    Answer
    = [8; 5]
  3. 4(1, 1, 0) - 2(0, 3, 1)
    Answer
    = [4; −2; −2]
  4. (1, 2, 3, 4) + (0, -1, 2, 5)
    Answer
    = [1; 1; 5; 9]
  5. 2(1, -1, 0, 3) - (3, 0, 2, 1)
    Answer
    = [−1; −2; −2; 5]
  6. -2(1, 3) + (5, 2)
    Answer
    = [3; −4]
  7. 3u - 2v for u = (1, 2, 0), v = (2, -1, 4)
    Answer
    = [−1; 8; −8]
  8. 2u + v for u = (3, -2), v = (-1, 5)
    Answer
    = [5; 1]
  9. 5(1, 2) - 2(3, -4)
    Answer
    = [−1; 18]

Harder

  1. u + 2v - w for u = (1, 0, 2), v = (0, 1, -1), w = (3, 1, 1)
    Answer
    = [−2; 1; −1]
  2. 2(1, 0, 1) + 3(0, 1, 1) - (1, 1, 0)
    Answer
    = [1; 2; 5]
  3. 3(2, 1, -2) - 2(3, 0, -1) + (0, 4, 1)
    Answer
    = [0; 7; −3]
  4. 2a - 3b for a = (1, -2, 0, 4), b = (2, -1, 1, 3)
    Answer
    = [−4; −1; −3; −1]
  5. (4, 2) - 2(2, 1)
    Answer
    = [0; 0]