We’ll start with a textbook definition of a basis:

A basis (for a nontrivial subspace $H$ of a vector space $V$) is an indexed set $\mathcal{B} = \{ v_1, …, v_p \}$ in $V$ such that:

  1. $\mathcal{B}$ is a linearly independant set
  2. $H = Span \{ v_1, …, v_p \}$

It should also be noted that any vector space $V$ is a subspace of itself, therefore, in the above definition, we can replace $H$ with $V$.

This definition is appropriate because of its rigor. But how can we strengthen our intuition about what a basis is?

TODO:

collection of coordinate systems in R^2

Quote From HN: a real matrix corresponded to a linear transformation of that space via a rotation, a reflection, a stretching, a shearing, etc.