Exercise 13

Show that this

is a linearly independent subset of $\mathbb{R}^{3}$.

Show that

is in the span of $S$ by finding $c_{1}$ and $c_{2}$ giving a linear relationship.

Show that the pair $c_{1},c_{2}$ is unique.

Assume that $S$ is a subset of a vector space and that $\vec{v}$ is in $[S]$, so that $\vec{v}$ is a linear combination of vectors from $S$. Prove that if $S$ is linearly independent then a linear combination of vectors from $S$ adding to $\vec{v}$ is unique (that is, unique up to reordering and adding or taking away terms of the form $0\cdot\vec{s}$). Thus $S$ as a spanning set is minimal in this strong sense: each vector in $[S]$ is a combination of elements of $S$ a minimum number of times — only once.

Prove that it can happen when $S$ is not linearly independent that distinct linear combinations sum to the same vector.