Let $T: \R^4 \to \R^2$ $ T: \mat{x_{1}\\x_{2}\\x_{3}\\x_{4}} \mapsto \mat{x_{1}-x_{2}+3x_{4}\\-1x_{1}+x_{2}+x_{3}+x_{4}} $ - Write a [[Algebraic Specification]] of the [[../02 Areas/Math/Kernel|Kernel]] of the [[../02 Areas/Math/Function|Transformation]] $T$ - Write a [[../02 Areas/Math/Set|Set]] of [[Vector|Vectors]] that [[../02 Areas/Math/Span|Span]] $\op{ker}(T)$ $ \let A = \mat{ 1&-1&0&3\\=1&1&1&1 } $ $ \begin{align} \op{ker}(T) = \op{Nul}(A) = \left\{ \vec x \in \R^n \left| T(\vec x) =0 \right. \right\} \end{align} $ $ \begin{align} &= \left\{ \vec x \in \R^4 \left| A\vec x = \vec 0 \right. \right\}\\ &= \boxed{ \left\{ \vec x \in \R^4 \left| \,\begin{split} x_{1}-x_{2}+3x_{4}&=0\\ -x_{1}+x_{2}+x_{3}+x_{4}&=0 \end{split} \right. \right\} }\\ \end{align} $