Extra (optional) exercises – Long Day#
These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.
Extra exercise 1: Isomorphic vector spaces#
Let \(V\) and \(W\) be vector spaces over a field \(\mathbb F\). \(V\) and \(W\) are called isomorphic if an invertible linear map \(L: V \to W\) exists.
Question a#
Let \(V\) be a finite-dimensional vector space, and let \(n=\dim_{\mathbb F}(V)\). Show that \(V\) and \(\mathbb F^n\) are isomorphic vector spaces.
Hint
With an ordered basis \(\beta\) for \(V\), any vector \(\mathbf v \in V\) can be associated with a \(\beta\)-coordinate vector \([\mathbf v]_{\beta} \in \mathbb F^n\).
Question b#
Let both \(V\) and \(W\) be finite-dimensional vector spaces. Show that \(V\) and \(W\) are isomorphic if \(\dim_{\mathbb F}(V)=\dim_{\mathbb F}(W)\).
In fact, \(V\) and \(W\) are isomorphic if and only if \(\dim_{\mathbb F}(V)=\dim_{\mathbb F}(W)\).
Extra exercise 2: The sum of subspaces#
Let \(V\) be an \(n\)-dimensional vector space over a field \(\mathbb{F}\), and let \(W_1\) and \(W_2\) be two subspaces of \(V\). The sum \(W_1+W_2\) of the two subspaces is defined as
Question a#
Show that \(W_1+W_2\) is also a subspace of \(V\).
Question b#
Show that if \(W_1 \cap W_2= \{ \mathbf 0\}\), then every vector \(\mathbf v\) in \(W_1+W_2\) can be written uniquely in the form \(\mathbf v = \mathbf w_1+\mathbf w_2\) for some \(\mathbf w_1 \in W_1, \mathbf w_2 \in W_2\).
Hint
If \(\mathbf w_1+\mathbf w_2=\tilde{\mathbf w}_1+\tilde{\mathbf w}_2\) for \(\mathbf w_1, \tilde{\mathbf w}_1 \in W_1\) and \(\mathbf w_2, \tilde{\mathbf w}_2 \in W_2\), then \(\mathbf w_1- \tilde{\mathbf w}_1 = -\mathbf w_2+ \tilde{\mathbf w}_2\).
Question c#
Show that if \(W_1 \cap W_2= \{ \mathbf 0\}\), then \(V=W_1+W_2\) if and only if \(\dim_{\mathbb F}(W_1)+\dim_{\mathbb F}(W_2)=\dim_{\mathbb F}(V)\).