Statement:
Let $V$ a vector subspace of $\mathbb{R}^4$. The bilinear form $f(x,y) = x_1y_1 + x_2y_2 + x_3y_3 - x_4y_4$ where $x,y \in \mathbb{R}$. Let $V^\bot = \{ y \in \mathbb{R}^4 : f(x, y) = 0 \ \ \ \forall x \in V \}$.
Prove that $ \text{dim}(V) + \text{dim}(V^\bot) = 4$.
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This problem currently had multiple subproblems which I've already solved successfully (e.g. prove that $V^\bot$ is a subspace, find $V$ that $ V \cap V^\bot \ne \{ 0 \} $ ).
I've tried giving a basis of $V \cap V^\bot$ and extending it to a basis of $V$ and a basis of $V^\bot$ but I couldn't go further.
I'll appreciate any hint. Thanks.