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: The expansion principle

Let \(\mathbf v_1,\dots,\mathbf v_n\) be vectors in \(\mathbb{F}^n\) and let \(V\) be defined as \(V=\mathrm{Span}(\mathbf v_1,\dots,\mathbf v_n)\). Let \(d=\dim(V)\), and assume that \(\mathbf w_1,\dots,\mathbf w_e\) are linearly independent vectors in \(V\), where \(e\) is an integer satisfying \(e \le d\).

Show that vectors \(\mathbf w_{e+1},\dots,\mathbf w_d\) exist in \(V\) such that \((\mathbf w_1,\dots,\mathbf w_d)\) is an ordered basis for \(V\). In other words, each set of linearly independent vectors in \(V\) can be expanded to form an ordered basis for \(V\).

Hint

The vectors \(\mathbf w_1,\dots,\mathbf w_e,\mathbf v_1,\dots,\mathbf v_n\) span \(V\). Can Theorem 9.2.1 be of use?

Extra exercise 2: Coset

Let \(\mathbf v_1,\dots,\mathbf v_n\) be vectors in \(\mathbb{F}^n\) and let \(V\) be defined as \(V=\mathrm{Span}\) \(\mathbf v_1,\dots,\mathbf v_n\). A coset of \(V\) by a vector \(\mathbf w \in \mathbb{F}^n\) is defined as follows:

\[\mathbf w + V =\{\mathbf w+\mathbf v \, | \, \mathbf v \in V \}.\]

Question a

Show that the solution set to an inhomogeneous system of linear equations is a coset of the solution set to the corresponding homogeneous system.

Question b

Show that \(\mathbf w_1 + V = \mathbf w_2 + V\) if and only if \(\mathbf w_1-\mathbf w_2 \in V\). Conclude that the difference between two particular solutions to an inhomogeneous equation system is a solution to the corresponding homogeneous equation system.