Let \(I\subset \mathbb{R}\) be a non-empty interval. Show that the set \(\mathsf{F}(I,\mathbb{K})\) of mappings from \(I\) into the field \(\mathbb{K}\) admits the structure of a \(\mathbb{K}\)-vector space.

Which of the following subsets of \(M_{n,n}(\mathbb{R})\) are vector subspaces?

1. the subset of symmetric matrices (i.e., those matrices \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) satisfying \(\mathbf{A}=\mathbf{A}^T\));

2. the subset of invertible matrices;

3. the subset of upper triangular matrices (i.e., those matrices \(\mathbf{A}=(A_{ij})_{1\leqslant i,j\leqslant n} \in M_{n,n}(\mathbb{R})\) satisfying \(A_{ij}=0\) for all \(i>j\)).

Justify your answer.

For \(n \in \mathbb{N}\cup \{0\}\) let \(g_n : \mathbb{R}\to \mathbb{R}\) be defined by \(x \mapsto x^n.\) Show that \(\{g_n\}_{n \in \mathbb{N}\cup\{0\}}\) is a basis for \(\mathsf{P}(\mathbb{R}).\)

Let \(U\) be the subset of \(\mathbb{R}^\infty\) consisting of the sequences \(x:\mathbb{N}\to\mathbb{R}\) satisfying \[x_{n+2}=x_{n+1}+x_n\] for all \(n\in\mathbb{N}.\) The Fibonacci sequence is the sequence \((x_n)_{n\in \mathbb{N}}\in U\) that satisfies \(x_1=x_2=1.\)

1. Show that \(U\) is a subspace of \(\mathbb{R}^\infty.\)

2. Let \(y_n = a_1^n\) and \(z_n=a_2^n,\) where \(a_1,a_2\in\mathbb{R}.\) Find \(a_1\) and \(a_2\) such that the set \(\{(y_n)_{n \in \mathbb{N}},(z_n)_{n\in \mathbb{N}}\}\) forms a basis of \(U.\)

3. Show Binet’s formula (1843): \[x_n = \frac{1}{\sqrt 5}\left(\left(\frac{1+\sqrt 5}{2}\right)^n-\left(\frac{1-\sqrt 5}{2}\right)^n\right).\]