Step by step
The question
Question: find the coordinates of (4, 7) in the basis (1, 1) and (1, 2).
We call the basis vectors b₁ and b₂, 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 and c₂.
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 and R 2.
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 = 1.
Row 2: c₂ = 3.
We write them as a column, in the basis order: (1, 3).
Careful!
The order follows the basis.
c₁ goes with b₁, so it comes first.
5 Check
Let's check: b₁ + 3b₂ must give v.
We get (4, 7): 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.