Step by step
The question
Question: make (1, 1) and (1, 0) orthonormal, with Gram-Schmidt.
We want new vectors u₁ and u₂, each at a right angle to the others, that make the same vectors as the old ones.
The idea: take away from each vector the parts along the earlier ones.
1 Keep the first
The first one stays: u₁ = v₁.
2 Make the second one perpendicular
u₂ is v₂ with its part along u₁ taken away.
Careful!
The projection is taken away from v₂, not added.
The part along u₁, as in the projection lesson.
The top: v₂ • u₁.
The bottom: u₁ • u₁.
So the number is 1/2.
Times u₁: every entry times 1/2.
The projection is (1/2, 1/2).
So u₂ is v₂ minus the projection.
Entry by entry: (1/2, −1/2).
To clear the fractions, we multiply it by 2.
Same direction, still at a right angle.
3 Divide by the lengths
Orthonormal means at right angles, and each one has length 1.
So we divide each one by its length.
The length of u₁: √2.
So e₁ is u₁ divided by √2.
The length of u₂: √2.
So e₂ is u₂ divided by √2.
So the orthonormal vectors are e₁ = 1 over √2 times (1, 1) and e₂ = 1 over √2 times (1, −1).
4 Check
Let's check: every pair must have dot product 0.
u₁ • u₂.
And each length is 1: the length of u₁ divided by itself.
Every dot product is 0: the vectors are at right angles.
So the answer is right!
Let's look back.
First: keep the first.
Then: make the second one perpendicular.
Then: divide by the lengths.
Last: check.