 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2011/2/a/solution

The Gram-Schmidt process subtracts projections onto directions already constructed and then normalizes. Define recursively

w_{k} = v_{k} - \sum_{j < k} ⟨ v_{k}, e_{j} ⟩ e_{j}, e_{k} = \frac{w _{k}}{∥ w _{k} ∥}, k = 1, \dots, n,

where the empty sum for

k = 1

is zero. Assuming the preceding

e_{j}

w_{k}

with

e_{i}

i < k

, gives zero. Moreover

w_{k} \neq = 0

: otherwise

v_{k}

would belong to the span of

v_{1}, \dots, v_{k - 1}

, contrary to linear independence. Normalization is therefore legitimate and produces an orthonormal basis. At each stage the first

k

constructed vectors have the same span as the first

k

original vectors.

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