Q1: Consider the subspace $U \subseteq \mathbb{R}^3$ spanned by the basis $\mathcal{B} = \Big( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \Big) $. What are the coordinates of the vector $\begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} $ with respect to the basis $\mathcal{B}$?

A1: $\lambda_1 = 0$ and $\lambda_2 = 1$

A2: $\lambda_1 = 2$ and $\lambda_2 = 1$

A3: $\lambda_1 = 1$ and $\lambda_2 = 1$

A4: $\lambda_1 = 4$ and $\lambda_2 = 5$

A5: $\lambda_1 = 0$ and $\lambda_2 = 3$

Q2: Consider the subspace $U \subseteq \mathbb{R}^3$ spanned by the basis $\mathcal{B} = \Big( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \Big) $. What are the coordinates of the vector $\begin{pmatrix} 3 \ 1 \ 3 \end{pmatrix} $ with respect to the basis $\mathcal{B}$?

A1: $\lambda_1 = 1$ and $\lambda_2 = 1$

A2: $\lambda_1 = 2$ and $\lambda_2 = 1$

A3: $\lambda_1 = 0$ and $\lambda_2 = 1$

A4: $\lambda_1 = 4$ and $\lambda_2 = 5$

A5: $\lambda_1 = 0$ and $\lambda_2 = 3$

Q3: Consider the subspace $U \subseteq \mathbb{R}^3$ spanned by the basis $\mathcal{B} = \Big( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \Big) $. What are the coordinates of the vector $\begin{pmatrix} 1 \ 5 \ 0 \end{pmatrix} $ with respect to the basis $\mathcal{B}$?

A1: $\lambda_1 = 5$ and $\lambda_2 = 1$

A2: $\lambda_1 = 2$ and $\lambda_2 = 1$

A3: $\lambda_1 = 1$ and $\lambda_2 = 1$

A4: $\lambda_1 = 1$ and $\lambda_2 = 5$

A5: $\lambda_1 = 0$ and $\lambda_2 = 3$