 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/18/b/i/solution

Take the dot product of the orthogonal basis expansion with

u_{j}

. All terms except the

j

th vanish by orthogonality, leaving

u_{j} \cdot x = α_{j} u_{j} \cdot u_{j} .

Therefore

α_{i} = \frac{u _{i} \cdot x}{u _{i} \cdot u _{i}} .

The basis is not assumed normalized, so its squared lengths must remain in the denominators. Applying

S

and using

S u_{i} = λ_{i} u_{i}

gives

S x = \sum_{i} λ_{i} α_{i} u_{i}, β_{i} = λ_{i} \frac{u _{i} \cdot x}{u _{i} \cdot u _{i}} .

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