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: Considering complex vector spaces as real vector spaces

Let \(V\) be a finite-dimensional complex vector space. We will write \(n=\dim_{\mathbb C}(V)\).

Question a

Confirm that \(V\) can be considered a real vector space since addition of vectors and the scalar-vector product \(c \cdot \mathbf v\) for \(c \in \mathbb{R}\) and \(\mathbf v \in V\) are defined by using the fact that \(V\) is a complex vector space.

Question b

Show that \(\dim_{\mathbb R}(V)=2\dim_{\mathbb C}(V)=2n\).

Hint

Let \((\mathbf v_1,\dots,\mathbf v_n)\) be an ordered basis for \(V\) considered as a complex vector space. Can you find an ordered basis for \(V\) considered as a real vector space?

Hint

Check that \((\mathbf v_1,i\cdot \mathbf v_1,\dots,\mathbf v_n,i\cdot \mathbf v_n)\) is an ordered basis for \(V\) considered as a real vector space.

Extra exercise 2: Complete flags

Let \(n\) and \(m\) be natural numbers, \(V\) an \(n\)-dimensional vector space over a field \(\mathbb{F}\) and \(W\) an \(m\)-dimensional subspace of \(V\).

Question a

Assume that \(m <n-1\). Show that a subspace \(\tilde{W}\) of \(V\) exists such that \(W\) is a subspace of \(\tilde{W}\).

Hint

Assume that \((\mathbf w_1,\dots,\mathbf w_m)\) is an ordered basis for \(W\). Use your answers to Extra exercise 1 from Long Day in Week 8.

Question b

A sequence of subspaces \(W_0,W_1,\dots,W_n\) of \(V\) that satisfy \(\dim(W_i)=i\) for all \(i\), as well as \(W_0 \subset W_1 \subset \cdots \subset W_n\), is called a complete flag in \(V\). Show that a complete flag exists in \(V\) such that \(W_m=W\).

Question c

Now let \(V\) be the real vector space \(\mathbb{R}^4\). Construct two complete flags \(V_0,V_1,V_2,V_3,V_4\) and \(W_0,W_1,W_2,W_3,W_4\) in \(\mathbb{R}^4\) such that \(\dim(V_2 \cap W_2)=0\) and \(\dim(V_3 \cap W_3)=2\). Is it possible to construct two complete flags in \(\mathbb{R}^4\) that satisfy \(\dim(V_2 \cap W_2)=0\) and \(\dim(V_3 \cap W_3)=3\)?