### 1.

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?

### 2.

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 $$\mathbf{u}=3+\frac{2}{7}\sqrt{11}\in V\text{.}$$

### 3.

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

### 4.

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

### 5.

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