(Solved): Subspaces: Question 1: Write any two non-zero elements from the given subsets ...

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Subspaces: Question 1: Write any two non-zero elements from the given subsets. (a). \( S_{1}=\{(a, b, c) \mid a+b=2+c\} \subseteq \mathbb{R}^{3} \). (b). \( S_{2}=\{(a, b, c, d) \mid a+d=b+c\} \subseteq \mathbb{R}^{4} \). (c). \( S_{3}=\left\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \mid c=b\right. \) and \( \left.a=d^{2}\right\} \subseteq M_{2 \times 2}(\mathbb{R}) \). Question 2: Decide which of the following subsets are subspaces of the given vector spaces. (a). \( W_{1}=\{(a, b) \mid a+b=0\} \subseteq \mathbb{R}^{2} \). (b). \( W_{2}=\{(a, 0, c) \mid a, c \in \mathbb{R}\} \subseteq \mathbb{R}^{3} \). (c). \( W_{3}=\{a+b x \mid a=b+1\} \subseteq P_{1}(x) \). (d). \( W_{4}=\left\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \mid a d-b c \neq 0\right\} \subseteq M_{2 \times 2}(\mathbb{R}) \). (e). \( W_{5}=\left\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \mid a+d=0\right\} \subseteq M_{2 \times 2}(\mathbb{R}) \)