Let $u,v,w \in \mathbb{R}^4$ and define the linear function by $F(x)=det[x \ u \ v \ w]$ for all $x \in \mathbb{R}^4$. Prove that there is a vector $z$ in $\mathbb{R}^4$ such that $T(x)=z \cdot x$ (dot product) for all $x$.
Please also find the components of $z$ in terms of $u,v,w$.
It makes sense to think about but I do not know where to start the proof, any tips or help would be greatly appreciated!
You just need to show that $F(\mathbf{x})$ is a linear function of $\mathbf{x}$ i.e. $F(k\mathbf{x}) = kF(\mathbf{x})$ and $F(\mathbf{x} + \mathbf{y}) = F(\mathbf{x}) + F(\mathbf{y})$