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I am familiar with finding the basis and dimension of simpler vector spaces but ones like the following are giving me some trouble; so I would appreciate some general hints/explanation on generating a basis for them:

  • For the field of reals $R$, let the vector space be a set of functions from $R\to R$ which are solutions to some differential equation (e.g. $2 \frac{d^2f}{dx^2} + 5f =0$)
  • For $R$ again, and a more standard vector space being the set of all vectors $(a, b, c)$ in $R^3$ such that $a + b - c = 0$ (I assume this means the standard basis is out of the question as it doesn't adhere to some condition placed on a,b,c)

Thanks!

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Hint: These are both subsets of some larger vectors space: the space of functions from $\mathbb R$ to $\mathbb R$ in the first case, and the $\mathbb R^3$. You don’t need to verify all of the vector space axioms to show that these subsets are subspaces because they will “inherit” many of the required properties from the parent space, but there are a few key ones that you do have to check. A key property to verify is closure under addition and scalar multiplication. There are a couple of others that need to be checked as well.

So, for instance, for the second subset, if you have two vectors $\mathbf v$ and $\mathbf w$ that both satisfy the equation, does their sum also satisfy it? Does a scalar multiple of $\mathbf v$ satisfy the equation? Similarly, if you have two functions that both satisfy the given differential equation, does their sum satisfy it?