Exercise 23

With a some calculation we can get formulas to determine whether or not a set of vectors is linearly independent.

• Show that this subset of $\mathbb{R}^{2}$

  $\{\hskip 1.00006pt\begin{pmatrix}a\\ c\end{pmatrix},\begin{pmatrix}b\\ d\end{pmatrix}\hskip 1.00006pt\}$

  is linearly independent if and only if $ad-bc\neq 0$.

• Show that this subset of $\mathbb{R}^{3}$

  $\{\hskip 1.00006pt\begin{pmatrix}a\\ d\\ g\end{pmatrix},\begin{pmatrix}b\\ e\\ h\end{pmatrix},\begin{pmatrix}c\\ f\\ i\end{pmatrix}\hskip 1.00006pt\}$

  is linearly independent iff $aei+bfg+cdh-hfa-idb-gec\neq 0$.

• When is this subset of $\mathbb{R}^{3}$

  $\{\hskip 1.00006pt\begin{pmatrix}a\\ d\\ g\end{pmatrix},\begin{pmatrix}b\\ e\\ h\end{pmatrix}\hskip 1.00006pt\}$

  linearly independent?

• This is an opinion question: for a set of four vectors from $\mathbb{R}^{4}$, must there be a formula involving the sixteen entries that determines independence of the set? (You needn’t produce such a formula, just decide if one exists.)