Step by step
The question
Question: are (1, 0) and (0, 1) linearly independent?
Independent means: the only way to make the zero vector from them is to take 0 of each.
Otherwise they are dependent.
1 Write the vector equation
We look for numbers c₁ and c₂ that make the zero vector.
Careful!
Taking 0 of each always gives 0.
That proves nothing.
The question is: is it the only way?
2 Write the augmented matrix
Each vector goes in as a column: the first entries make row 1, the second entries row 2.
After the bar: 0s.
Every move keeps them 0.
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.
The matrix already has its leading 1s, with 0s above and below.
No moves are needed.
4 Count the leading 1s
The first number that is not 0 in each row is a 1.
We call it a leading 1, and ring it.
Every column has a leading 1.
So each row says one c is 0: c₁ = c₂ = 0.
That is the only way.
So the vectors are linearly independent.
5 Check
Let's check another way.
The vectors make a square matrix.
Its determinant must not be 0.
So D = 1 × 1 − 0 × 0.
1 times 1 is 1.
0 times 0 is 0.
1 minus 0 is 1.
So D = 1.
D is 1, not 0.
So the vectors are independent: the answer is right!
Let's look back.
First: write the vector equation.
Then: write the augmented matrix.
Then: row reduce.
Then: count the leading 1s.
Last: check.