(Solved): Vector Spaces, Subspaces, Basis and Dimension Q1. Determine whether the given set is a vector space ...

Vector Spaces, Subspaces, Basis and Dimension Q1. Determine whether the given set is a vector space with the given defined addition and scalar multiplication operations. If it is a vector space, verify the vector space axioms; if not, state at least one vector space axiom that is not satisfied. a) The set of all pairs of real numbers of the form \( (x, 0) \) with the standard operations. b) The set of all pairs of real numbers of the form \( (x, y) \), where \( x \geq 0 \), with the standard operations. c) The set of all \( 2 \times 2 \) matrices of the form \( \left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right) \). d) \( V=\{(x, y, 1), x, y \in R\} \) \[ \begin{array}{l} \left(x_{1}, y_{1}, 1\right)+\left(x_{2}, y_{2}, 1\right)=\left(x_{1}+x_{2}, y_{1}, y_{2}+1\right) \\ c(x, y, 1)=(c x, c y, 1) \end{array} \] Q2. Check whether \( W \) is a subspace of \( R^{3} \) or not; \( W=\{(x, 2 x+y, 0): x \in R, y \geq 0\} \). Q3. Let \( A \) be a given \( 2 \times 3 \) matrix. Prove that the set: a) \( W=\left\{X \in R^{3}: A X=\left[\begin{array}{ll}1 & 2\end{array}\right]^{t}\right\} \) is not a subspace of \( R^{3} \). b) \( W=\left\{X \in R^{3}: A X=\left[\begin{array}{ll}0 & 0\end{array}\right]^{t}\right\} \) is a subspace of \( R^{3} \). Q4. Determine whether the set \( S \) is linearly independent or linearly dependent: a) \( S=\left\{2, x+3,3 x^{2}\right\} \) b) The set of matrices \( \left\{\left[\begin{array}{cc}1 & -1 \\ 4 & 5\end{array}\right],\left[\begin{array}{cc}4 & 3 \\ -2 & 3\end{array}\right],\left[\begin{array}{cc}1 & -8 \\ 2 & 2\end{array}\right]\right\} \) from \( M_{2,2} \). Q5. Check whether \( W ; W=\{(x, 2 x+y, 0): x \in R, y \geq 0\} \); is a subspace of \( R^{3} \) ? If it is a subspace, find its basis and dimension.