Proof of Theorem MLTLT

We simply check the defining properties of a linear transformation (Definition LT). This is another good place to consistently ask yourself which objects are being combined with which operations. \begin{align*} \lt{(\alpha T)}{\vect{x}+\vect{y}}&= \alpha\left(\lt{T}{\vect{x}+\vect{y}}\right)&&\knowl{./knowls/definition.LTSM.knowl}{\text{Definition LTSM}}\\ &=\alpha\left(\lt{T}{\vect{x}}+\lt{T}{\vect{y}}\right)&&\knowl{./knowls/definition.LT.knowl}{\text{Definition LT}}\\ &=\alpha\lt{T}{\vect{x}}+\alpha\lt{T}{\vect{y}}&&\knowl{./knowls/property.DVA.knowl}{\text{Property DVA}}\text{ in }V\\ &=\lt{(\alpha T)}{\vect{x}}+\lt{(\alpha T)}{\vect{y}}&&\knowl{./knowls/definition.LTSM.knowl}{\text{Definition LTSM}}\\ \end{align*} and \begin{align*} \lt{(\alpha T)}{\beta\vect{x}}&= \alpha\lt{T}{\beta\vect{x}}&&\knowl{./knowls/definition.LTSM.knowl}{\text{Definition LTSM}}\\ &=\alpha\left(\beta\lt{T}{\vect{x}}\right)&&\knowl{./knowls/definition.LT.knowl}{\text{Definition LT}}\\ &=\left(\alpha\beta\right)\lt{T}{\vect{x}}&&\knowl{./knowls/property.SMA.knowl}{\text{Property SMA}}\text{ in }V\\ &=\left(\beta\alpha\right)\lt{T}{\vect{x}}&&\text{Commutativity in $\complex{}$}\\ &=\beta\left(\alpha\lt{T}{\vect{x}}\right)&&\knowl{./knowls/property.SMA.knowl}{\text{Property SMA}}\text{ in }V\\ &=\beta\left(\lt{(\alpha T)}{\vect{x}}\right)&&\knowl{./knowls/definition.LTSM.knowl}{\text{Definition LTSM}}\\ \end{align*}