Question: Vectors are objects that you can add together; show that the set of all linear transformations mapping \(\mathbb{R}^{3} \rightarrow \mathbb{R}\) is itself a vector space. Find a basis for this vector space. Do you think your proof could be modified to work for linear transformations \(\mathbb{R}^{n} \rightarrow \mathbb{R} ?\) For \(\mathbb{R}^{\mathbb{N}} \rightarrow \mathbb{R}^{m} ?\) For \(\mathbb{R}^{\mathbb{R}} ?\)

Hint: Represent \(\mathbb{R}^{3}\) as column vectors, and argue that a linear transformation \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}\) is just a row vector.

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