MAS400: Exercises 8

Let v_{1}, …, v_{n} be linearly independent vectors in ℝ^{n}, such that (v_{1}, …, v_{n}) has Gram–Schmidt orthogonalisation ((v_{1}^{∗}, …, v_{n}^{∗}), (μ_{st})). Determine the Gram–Schmidt orthogonalisations ((w_{1}^{∗}, …, w_{n}^{∗}), (ξ_{st})) of (w_{1}, …, w_{n}) in the following cases.

1. w_{i} = λv_{i}, w_{j} = v_{j} if j ≠ i, where λ ≠ 0 (λ ∈ ℝ).

2. w_{i} = v_{i} − λv_{k} with k < i, w_{j} = v_{j} if j ≠ i.

3. w_{i} = v_{i+1}, w_{i+1} = v_{i}, w_{j} = v_{j} if j ≠ i, i + 1.

You may like to think about what happens in these two cases too. (It is not really possible to write down exact answers for these.)

4. w_{i} = v_{i} − λv_{k} with k > i, w_{j} = v_{j} if j ≠ i.

5. w_{i} = v_{k}, w_{k} = v_{i} (wlog i < k), w_{j} = v_{j} if j ≠ i, k.