Notes
Gram-Schmidt orthonormalization is a process for finding an orthonormal basis for a vector space. The resulting vectors are orthogonal (with inner product 0) and are unit vectors (with length/norm 1). Consider a basis for R2, consisting of two vectors that need not be orthogonal nor unit vectors: B = {v1, v2}.
- An orthogonal basis for R2 is B’ = {w1, w2}, where w1 = v1 and w2 = v2 – Projw1(v2) = v2 – (v2·w1 / w1·w1) w1.
- An orthonormal basis for R2 is B” = {u1, u2}, where u1 = w1 / ||w1|| and u2 = w2 / ||w2||.
For example, suppose B = {v1, v2} = {(3, 4)T, (1, 0)T}.
- Then, w1 = v1 = (3, 4)T,
- and w2 = v2 – (v2·w1 / w1·w1) w1 = (1, 0)T – (3/25) (3, 4)T = (16/25, –12/25)T.
- Next, ||w1|| = √(32+42) = √(9+16) = √25 = 5,
- and ||w2|| = (1/25) √(162+122) = (1/25) √(256+144) = (1/25) √400 = 20/25 = 4/5.
- Finally, u1 = w1 / ||w1|| = (3/5, 4/5)T,
- and u2 = w2 / ||w2|| = (4/5, –3/5)T.
- Check u1·u2 = 0 and ||u1|| = ||u2|| = 1.
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Practice Exercises
For the following bases in R2, apply the Gram-Schmidt orthonormalization process to the vectors in the basis (in the order given). Write all numbers as decimals (not fractions).
