Formal system of vector spaces Theorems in book Settle on axioms. Then see how far they lead you. First: subspace: hierarchy of formal systems: set theory: ZFC: built formal system of fields; formal system of vector spaces is built on that. V S references specific field. Notion of sub, something included. Subset of S, some but not necessarily all of S Then, intersection $\cap$ $ and union $\cup$ Structure introduced to sets Field: set; two operations: + $\cdot$ consider subsets compatible with structures of field; G $\subset$ F G is equipped with operations above; Field of real numbers R $\mathbb{R}$ $\subset$ $\mathbb{C}$ VS subspace : V is vector space over field F subset U of V is called subspace of V if it is preserved by operations of addition and scalar multiplication $\forall$ v, w $\in$ U UXU to U, FXU to U U with these operations satisfies VS Same F as original F for defining V Think of $\mathbb{R}^2$ as collection of column vectors, pairs of real numbers What are subspaces? U with colon before equal: := U := 1 0 U y1 y2 Theorem: define subspace 0 $\in$ U u + v $\in$ U lambda times u $\in$ U