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Past exam of the mathematics course of the University of Oxford

2021

Prelims

Mathematics I

{c}

<Bilinearity> of the real <inner product> gives, for the specified <linear combinations>,
$$
\begin{aligned}
\langle u,v\rangle
&=\left\langle\sum_{i=1}^n x_iv_i,\sum_{j=1}^n y_jv_j\right\rangle\\
&=\sum_{i=1}^n\sum_{j=1}^n x_i y_j\langle v_i,v_j\rangle
=\sum_{i=1}^n\sum_{j=1}^n x_iA_{ij}y_j
=\boxed{x^TAy}.
\end{aligned}
$$
The last equality is the entrywise rule for <matrix multiplication>, with $x^T$ the <transpose> of the column <vector> $x$.

Solution

Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 4 / b / i / Solution

Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 4 / b / i / Solution

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Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 4 / b / i / Solution

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