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Conditionality Statements

Translated into the if the antecedent $P$, then consequent $Q$ structure, we write our theorem propositions

Proof Intuition

Notice:

• Distributivity and Scalar Multiplication $⟹$ Linearity
• Linearity $⟹$ Distributivity and Scalar Multiplication

Intuitive draft: If distributivity and scalar multiplication hold, then linearity must be true.

We can translate our intuitive draft into a technical draft of conditionality as follows:

Proofs

To prove our theorems, let's make a conjecture.

Let's show that conjecture 1 is true.
\begin{proof}
Let $A\in {\mathbb{R}}^{m\times n},\overrightarrow{x},\overrightarrow{y}\in {\mathbb{R}}^{m\times 1},\text{ and }\alpha ,\beta \in \mathbb{R}$. Suppose distributivity and scalar multiplication hold true for any vectors and scalars.

Consider

This is what we wanted to show.
\end{proof}
Now, we have to prove conjecture 2.
\begin{proof}
Let $A\in {\mathbb{R}}^{m\times n},\overrightarrow{x},\overrightarrow{y}\in {\mathbb{R}}^{m\times 1},\text{ and }\alpha ,\beta \in \mathbb{R}$. Suppose linearity holds true for any vectors and scalars. First, let's prove distributivity. Consider

This proves distributivity. Next, we must prove scalar multiplication. Consider

\end{proof}
These proofs show that linearity $⟺$ distributivity and scalar multiplication. Finally, let's prove linearity using the 10.4 Row-Vector-Matrix Multiplication via Linear Combinations definition.
\begin{proof}
Let $A\in {\mathbb{R}}^{m\times n},\overrightarrow{x},\overrightarrow{y}\in {\mathbb{R}}^{m\times 1},\text{ and }\alpha ,\beta \in \mathbb{R}$. Consider

This is what we wanted to show.
\end{proof}