Orthonormal basis from (3, 4), (1, 0)

Step by step

The question

Question: make (3, 4) 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 3/25.

Times u₁: every entry times 3/25.

The projection is (9/25, 12/25).

So u₂ is v₂ minus the projection.

Entry by entry: (16/25, −12/25).

To clear the fractions, we multiply it by 25/4.

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: 5.

So e₁ is u₁ divided by 5.

The length of u₂: 5.

So e₂ is u₂ divided by 5.

So the orthonormal vectors are e₁ = 1/5 times (3, 4) and e₂ = 1/5 times (4, −3).

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.