What can you build from two directions?

Source scope: §3.1. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Build a target

Suppose two adjustable ingredients contribute $u=(1,0)$ and $v=(1,2)$. To build $(3,2)$, take two copies of $u$ and one of $v$: $2u+v=(3,2)$. The coefficients may be negative or zero. We are describing combinations, not quantities you must buy.

A real vector space is a set $V$ with addition $V\times V\to V$ and scalar multiplication $\mathbb R\times V\to V$. These operations stay in $V$. For every $x,y,z\in V$ and $\alpha,\beta\in\mathbb R$, the source lists these axioms:

1. An additive identity $0\in V$ satisfies $x+0=x$.
2. Each $x$ has an additive inverse $-x$, with $x+(-x)=0$.
3. The scalar identity satisfies $1x=x$.
4. Addition commutes: $x+y=y+x$.
5. Addition and scaling associate: $(x+y)+z=x+(y+z)$ and $\alpha(\beta x)=(\alpha\beta)x$.
6. Both distributive laws hold: $\alpha(x+y)=\alpha x+\alpha y$ and $(\alpha+\beta)x=\alpha x+\beta x$.

Source footnote 1: Vector spaces can use any field of scalars. These notes use the real field to avoid a detour into abstract algebra.

Reach and redundancy

A linear combination is $\sum_{i=1}^k\alpha_i v_i$. The span is the set of all such combinations:

The vectors are linearly independent when

Our two ingredients are independent: the second coordinate of $au+bv=0$ gives $2b=0$, then the first gives $a=0$. They also span the plane: $(s,t)=(s-t/2)u+(t/2)v$. A basis is both independent and spanning. Every independent list is a basis for its own span.

If a finite list spans $V$, the space is finite-dimensional. Its dimension $\dim V$ is the number of vectors in a basis. Otherwise it is infinite-dimensional. As a teaching example, all real polynomials form an infinite-dimensional space: no finite list can generate polynomials of arbitrarily high degree.

Transfer

Can $(1,2)$ and $(2,4)$ form a basis for the plane?