Abstract Algebra: Theory and Applications

Why do the axioms of a vector space appear to only have four conditions, rather than the ten you may have seen the first time you saw an axiomatic definition?

The set $V=\mathbb{Q}(\sqrt{11})=\{a+b\sqrt{11}\mid a,b\in \mathbb{Q}\}$ is a vector space. Carefully define the operations on this set that will make this possible. Describe the subspace spanned by $S=\{\mathbf{u}\}\text{,}$ where $\mathbf{u}=3+\frac{2}{7}\sqrt{11}\in V\text{.}$

Write a long paragraph, or a short essay, on the importance of linear independence in linear algebra.

Write a long paragraph, or a short essay, on the importance of spanning sets in linear algebra.

“Linear algebra is all about linear combinations.” Explain why you might say this.