- #1

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If [itex]V[/itex] is a subspace of [itex]\Re^{n}[/itex], prove that [itex]V^{\bot}[/itex] is also a subspace.

As usual, this is not homework. I am just a struggling hobbyist trying to better myself on my own time.

The only progress I've been able make towards a formal proof is to unpack the definition of a subspace: a set of objects which is closed over two operations: [itex]o_{1} + o_{2}[/itex] and [itex]a \cdot o[/itex], where [itex]o, o_{1}[/itex], and [itex]o_{2} [/itex] are any objects in the set, and a is a real number.

So what I want to show is that for two objects [itex]o_{1}, o_{2} \in V^{\bot}[/itex], their sum [itex]o_{1} + o_{2} \in V^{\bot}[/itex], and [itex]a \cdot o \in V^{\bot}[/itex].

But, I think this is the sticking point: what does it take to formally establish that some vector [itex]o[/itex] is in [itex]V^{\bot}[/itex]?