Step by step
The question
Question: find the coordinates of (0, 0, 6) in the basis (1, 1, 1), (1, −1, 0) and (1, 1, −2).
We call the basis vectors b₁, b₂ and b₃, in this order, and the vector v.
The coordinates of v are the numbers that make v from the basis vectors, in the basis order.
1 Write the vector equation
We look for c₁, c₂ and c₃.
2 Write the augmented matrix
Each vector goes in as a column: the first entries make row 1, the second entries row 2.
v goes after the bar.
We call the rows R 1, R 2 and R 3.
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.
4 Read the coordinates
Each row is an equation again.
Row 1: c₁ = 2.
Row 2: c₂ = 0.
Row 3: c₃ = −2.
We write them as a column, in the basis order: (2, 0, −2).
Careful!
The order follows the basis.
c₁ goes with b₁, so it comes first.
5 Check
Let's check: 2b₁ − 2b₃ must give v.
We get (0, 0, 6): that is v.
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 coordinates.
Last: check.