Notes
Consider a basis for R3, consisting of three vectors that need not be orthogonal nor unit vectors: B = {v1, v2, v3}.
- An orthogonal basis for R3 is B’ = {w1, w2, w3}, where
- w1 = v1,
- w2 = v2 – Projw1(v2) = v2 – (v2·w1 / w1·w1) w1,
- w3 = v3 – Projw1(v3) – Projw2(v3) = v3 – (v3·w1 / w1·w1) w1 – (v3·w2 / w2·w2) w2.
- An orthonormal basis for R2 is B” = {u1, u2, u3}, where u1 = w1 / ||w1||, u2 = w2 / ||w2||, and u3 = w3 / ||w3||.
For example, suppose B = {v1, v2, v3} = {(0, 1, 1)T, (1, 1, 0)T, (1, 0, 1)T}.
- Then, w1 = v1 = (0, 1, 1)T,
- and w2 = v2 – (v2·w1 / w1·w1) w1 = (1, 1, 0)T – (1/2) (0, 1, 1)T = (1, 1/2, –1/2)T.
- and w3 = v3 – (v3·w1 / w1·w1) w1 – (v3·w2 / w2·w2) w2 = (1, 0, 1)T – (1/2) (0, 1, 1)T – (1/3) (1, 1/2, -1/2)T = (2/3, -2/3, 2/3)T.
- Next, ||w1|| = √(12+12) = √(1+1) = √2,
- and ||w2|| = (1/2) √(22+12+12) = (1/2) √(4+1+1) = √6/2.
- and ||w3|| = (1/3) √(22+22+22) = (1/3) √(4+4+4) = 2√3/3.
- Finally, u1 = w1 / ||w1|| = (0, √2/2, √2/2)T,
- and u2 = w2 / ||w2|| = (√6/3, √6/6, –√6/6)T.
- and u3 = w3 / ||w3|| = (√3/3, –√3/3, √3/3)T.
- Check u1·u2 = u1·u3 = u2·u3 = 0 and ||u1|| = ||u2|| = ||u3|| = 1.
Video Tips
Practice Exercises
For the following bases in R3, apply the Gram-Schmidt orthonormalization process to the vectors in the basis (in the order given). Write all numbers as decimals (not fractions) and round to 3 decimal places if necessary.
