Step by step
The question
Question: is (7, 3, 1) in the span of (2, 1, 0) and (1, 1, 1)?
We call them v₁ and v₂, and the vector we test b.
The span of some vectors is every linear combination of them: any numbers times them, added.
1 Write the vector equation
b is in the span if some numbers c₁ and c₂ make this true.
We look for them.
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.
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.
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 answer
Look at row 3: 0s, then 2 after the bar.
As an equation it says 0c₁ + 0c₂ = 2.
The left side is always 0, so this is false for every choice of numbers.
So no numbers work: (7, 3, 1) is not in the span.
5 Check
Let's check.
The other rows give c₁ = 2 and c₂ = 1.
They are the only numbers that fit those rows.
We try them.
We get (5, 3, 1), not (7, 3, 1).
So b is not in the span: 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.