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I was given this problem,

Consider the elements $f,g,h \in \mathbb Q^\mathbb Q$ defined by $f: t\to t-1,\ g: t \to t+1$ and $h: t \to t^2+1$. Then does the function $t \to t^2$ belong to $\mathbb Q\{f,g,h\}$?

How do you go about thinking about this kind of stuff.

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    what does $\mathbb{Q}\{f,g,h\}$ mean? – 2017-02-07
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    @dave presumably, the span of $f,g,h$ as a $\Bbb Q$-vector space – 2017-02-07
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    So you could try $h-\frac{1}{2}g+\frac{1}{2}f$ – 2017-02-07

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For better or worse, here is my way of thinking about it:

Represented as vectors in $\mathbb{Q}\{1,t,t^2\}$ (where we are considering these as functions $t \mapsto 1$, $t \mapsto t$, and $t \mapsto t^2$, resp.), you have $$f \simeq \pmatrix{-1\\1\\0}, \quad g \simeq \pmatrix{1\\1\\0}, \quad h \simeq \pmatrix{1\\0\\1}.$$ You want to know if you can linearly combine these to equal $\pmatrix{0\\0\\1}$ with rational coefficients.