Q1: Consider the subspace $U \subseteq \mathbb{R}^3$ defined by $U = \mathrm{Span}( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix} )$. What is not a basis of $U$?

A1: The family given by $ \left( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 3 \end{pmatrix} \right) $

A2: The family given by $ \left( \begin{pmatrix} 4 \ 2 \ 6 \end{pmatrix} \right) $

A3: The family given by $ \left( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix} \right) $

A4: The family given by $ \left( \frac{1}{5} \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix} \right) $

Q2: Consider the subspace $U \subseteq \mathbb{R}^3$ defined by $U = \mathrm{Span} \left( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \right)$. What is a basis of $U$?

A1: The family given by $ \left( \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} -1 \ 1 \ 0 \end{pmatrix} \right) $

A2: The family given by $\left( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \right)$

A3: The family given by $ \left( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \right) $

A4: The family given by $ \left( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix} \right) $

Q3: Let $(\mathbf{u}, \mathbf{v})$ be a family of vectors in $\mathbb{R}^3$ that are linearly independent. Which claim is correct?

A1: They form a basis of a subspace in $\mathbb{R}^3$.

A2: They form not a basis of a subspace in $\mathbb{R}^3$.

A3: They form a basis of $\mathbb{R}^3$.

A4: One needs more information.