Is (0, 2, 4) a linear combination of (1, 1, 1), (1, 0, -1) Step by Step

Step by step

The question

Question: is (0, 2, 4) a linear combination of (1, 1, 1) and (1, 0, −1)?

v₁ = [1; 1; 1], v₂ = [1; 0; −1]

We call them v₁ and v₂, and the vector we test b.

b = [0; 2; 4]

The span of some vectors is every linear combination of them: any numbers times them, added.

span: every c₁v₁ + c₂v₂

1 Write the vector equation

b is in the span if some numbers c₁ and c₂ make this true.

We look for them.

c₁[1; 1; 1] + c₂[1; 0; −1] = [0; 2; 4]

2 Write the augmented matrix

Each vector goes in as a column: the first entries make row 1, the second entries row 2.

b goes after the bar.

= [1, 1, 0; 1, 0, 2; 1, −1, 4]

We call the rows R 1, R 2 and R 3.

Careful!

Each vector is a column of the matrix, not a row.

Row 1 holds the first entries of v₁ and v₂, and of b.

Careful: the vectors are the COLUMNS

3 Row reduce

We use the three row moves: swap two rows, times a row by a number that is not 0, add a multiple of one row to another.

They never change the answers.

Our goal: a leading 1 in each row, with 0s above and below it.

swap two rows

× a row by a number (not 0)

add a multiple of one row to another

→ [1, 1, 0; 0, −1, 2; 0, −2, 4]

→ [1, 1, 0; 0, 1, −2; 0, −2, 4]

→ [1, 0, 2; 0, 1, −2; 0, 0, 0]

4 Read the answer

Each row is an equation again.

Row 1 says c₁ = 2.

c₁ = 2

Row 2 says c₂ = −2.

c₂ = −2

So yes: b = 2v₁ − 2v₂.

b is in the span.

b = 2v₁ − 2v₂

5 Check

Let's check: we work out 2v₁ − 2v₂.

= 2 × [1; 1; 1] − 2 × [1; 0; −1]

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

= [2; 2; 2] − [2; 0; −2]

= [2 − 2; 2 − 0; 2 − (−2)]

= [0; 2; 4]

We get (0, 2, 4): that is b.

So the answer is right!

Let's look back.

First: write the vector equation.

Then: write the augmented matrix.

Then: row reduce.

Then: read the answer.

Last: check.

Answer