[Math] Proving/disproving that a set is a vector space

linear algebravector-spaces

I'm having trouble interpretation some equations.

To prove whether a given set is a vector space, I know there are ~10 axioms I need to test, but I'm having trouble interpreting the equations to be able to test the axioms.

Here is one of the equations:

$$V=\big\{x=\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\in \mathbb{R}^3;x_1+x_2+x_3=0 \big\}$$

How do I apply the axioms to this set. Any hints would be great.

EDIT: I think this is a subset of $\mathbb{R}^3$ if I'm reading this correctly, so I guess I should be able to just test if this is a subset.
This would mean I need to test:

  1. The zero vector of $\mathbb{R}^3$ is in V
  2. V is closed under addition
  3. H is closed under multiplication by scalars

Is this the right way to go about this? Would I just plug in for the $x_n$ terms?

Best Answer

Your observation is indeed correct: Since $V\subset\mathbb{R}^3$ and we know $\mathbb{R}^3$ is a vector space, we know that it suffices to check that $V$ is a subspace, i.e.

  1. $V$ is nonempty,
  2. $V$ is closed under addition, and
  3. $V$ has closed under scalar multiplication.

Doug M shows why these hold in his answer. The idea: Pick arbitrary vectors of $V$ and show that they satisfy the three properties. What does an arbitrary vector of $V$ look like?

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