20.4: Reading Questions
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.