# Reducing the span of vectors

Out of interest what would be the best way to describe the spanning set of vectors a a and b

a=(0,3,-2)

b=(1,0,0).

Apart form a and b, what other vector belongs to the spanning set?

Do i have to reduce the vectors using Gauss or should i approach this differently?

Thanks!

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You could describe it geometrically as a certain plane. Algebraically, it's just vectors of the form $(\beta, 3\alpha,-2\alpha)$ (the span is the set of linear combinations of the vectors; so vectors of the form $\alpha(0,3,-2)+\beta(1,0,0)=(\beta, 3\alpha,-2\alpha)$). – David Mitra May 13 '12 at 19:38

The span of $a$ and $b$ is the set of linear combinations of the vectors $a$ and $b$. This isn't too hard to write down here; no need to reduce, the vectors have disjoint support. The span is all vectors of the form $$\alpha(0,3,-2)+\beta(1,0,0)=(\beta, 3\alpha,-2\alpha)$$ where $\alpha$ and $\beta$ are scalars.
In particular, taking $\alpha=1=\beta$ shows that $(1,3-2)$ is in the span. For any values of $\alpha$ and $\beta$, you have an element in the span.
In fact, the span is a two dimensional subspace of $\Bbb R^3$; so you could describe the span of $a$ and $b$ geometrically as a certain plane. Namely, the plane through the origin in $\Bbb R^3$ that contains both the vectors $a$ and $b$.