A Linear Equation System is Homogenous iff its constants are 0

Last updated Nov 1, 2022

Let $V, W$ be Vector Spaces over the same Field $F$. Consider the Linear Equation System $$T_{\alpha}(\mathbf{x}) = \mathbf{b}{\alpha} \text{ }\forall \alpha \in A$$ Then this system is a Homogenous Linear System If and Only If $\mathbf{b}{\alpha} = \mathbf{0} \in W$ $\forall \alpha \in A$.

TODO - cleanup, but idea is good $(\Rightarrow)$ $S$ is Solution Set and it is a Vector Subspace of $V$. Then $\mathbf{x} + \mathbf{y} \in S$ and $\forall \alpha \in A$ $$\begin{align*} \mathbf{b}{\alpha} &= T{\alpha}(\mathbf{x} + \mathbf{y})\\ &=T_\alpha(\mathbf{x}) + T_{\alpha}(\mathbf{y})\\ &=\mathbf{b}{\alpha}+ \mathbf{b}\alpha\\ \Rightarrow \mathbf{0} &= \mathbf{b}_{\alpha} \end{align*}$$ $\checkmark$

$(\Leftarrow)$: Establish that $S$ is a Vector Subspace by A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition. $\checkmark$