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Chapter 1: Problem 7
Write the \(y\)-axis as a line in \(\mathbb{R}^{2}\).
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Prove Theorem 1.5, part d: \((-1) \mathbf{v}=-\mathbf{v}\). You may want to use the strategy employed in one of the proofs of Theorem 1.3. Alternatively, just show that \(\mathbf{v}+((-1) \mathbf{v})=\mathbf{0}\) and use Theorem \(1.3\) itself.
Determine whether \(S=\left\\{(x, y) \in \mathbb{R}^{2} \mid x\right.\) and \(y\) are integers \(\\}\) is a subspace of \(R^{2}\).
Determine whether we obtain a vector space from \(\mathbb{R}^{2}\) with operations defined by $$ \begin{aligned} \left(v_{1}, v_{2}\right)+\left(w_{1}, w_{2}\right) &=\left(v_{1}+w_{1}-1, v_{2}+w_{2}\right) \\ r\left(v_{1}, v_{2}\right) &=\left(r v_{1}-r+1, r v_{2}\right) \end{aligned} $$
Determine whether we obtain a vector space from \(\mathbb{R}^{2}\) with operations defined by $$ \begin{aligned} \left(v_{1}, v_{2}\right)+\left(w_{1}, w_{2}\right) &=\left(v_{1}+w_{1}, v_{2}+w_{2}\right), \\ r\left(v_{1}, v_{2}\right) &=(0,0) . \end{aligned} $$ It is especially interesting to have an example where exactly one of the axioms fails. This indicates that such an axiom is essential in the sense that there is no way to derive it from the other axioms.
Write the set \(\left\\{\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right] \in \mathbb{M}(2,2) \mid a, b \in \mathbb{R}\right\\}\) as a plane in \(M(2,2)\).
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