Return
- I the rank of \( v_{1}, v_{2}, \ldots, v_{m} \) is at most \( n \)
\( \overrightarrow{v_{1}}, \overrightarrow{v_{2}}, \ldots, \overrightarrow{v_{m}} \) is always a basis of span \( \left(\overrightarrow{v_{1}}, \overrightarrow{v_{2}}, \ldots, \overrightarrow{v_{m}}\right) \)
( \( \overrightarrow{v_{1}}, \overrightarrow{v_{2}}, \ldots, \overrightarrow{v_{m}} \) must be linearly dependent if \( m>n \).
4
10 points
Let \( A=\left(\begin{array}{cccc}2 & 4 & -1 & 3 \\ 6 & a & -3 & 9 \\ -2 & -4 & 1 & -3\end{array}\right) \) and \( \vec{b} \in \mathbb{R}^{3} \). Select the combination(s) of \( a \) and \( \vec{b} \) which results in \( V=\left\{\vec{x} \in \mathbb{R}^{4} \mid A \vec{x}=\vec{b}\right\} \) being a subspace of \( \mathbb{R}^{4} \) with
\( \operatorname{dim}(V)=2 \)
Note: Select all valid values for \( a \) and \( \vec{b} \) separately.
\( \checkmark \)
\[
\begin{array}{l}
\vec{b}=\overrightarrow{0} \\
\vec{b} \neq \overrightarrow{0}
\end{array}
\]
\( \checkmark \)
\[
\begin{array}{l}
a=12 \\
a \neq 12
\end{array}
\]
5
10 points
Consider the linear map \( T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} \) such that for every \( \vec{v}=\left(\begin{array}{c}v_{1} \\ v_{2} \\ v_{3}\end{array}\right) \in \mathbb{R}^{3}, T(\vec{v})=\binom{2 v_{1}-v_{3}}{v_{2}+v_{3}} \). Find \( A \), the associated matrix of \( T \) where \( T \vec{v}=A \vec{v} \).
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