Catalog of theorems from Linear Algebra alongside proofs for each.

Theorem 1. Cancellation Law for Vector Addition (Thm 1.1)
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If x,y,z∈V such that x+z=y+z, then x=y.\text{If } x, y, z \in V \text{ such that } x + z = y + z\text{, then } x = y.

Corollary 2. Uniqueness of the Zero Vector (Cor. 1 to Thm 1.1)
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0∈V is unique.0 \in V \text{ is unique.}

Corollary 3. The Additive Inverse is Unique (Cor. 2 to Thm 1.1)
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y∈V that satisfy (VS 4)‾ are unique.y \in V \text{ that satisfy \underline{\href{/notes/linear-algebra/definitions#vs-4}{\text{(VS 4)}}} are unique.}

Theorem 4. Thm 1.2
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Let V be a vector space. Then the following are true:(a) 0x=0  ∀x∈V(b) (−a)x=−(ax)=a(−x)  ∀a∈F and x∈V(c) a0=0  ∀a∈F\text{Let } V \text{ be a vector space. Then the following are true:} \\[0.5em] \text{(a) } 0x = 0 \ \ \forall x \in V \\[0.5em] \text{(b) } (-a)x = -(ax) = a(-x) \ \ \forall a \in \mathbb{F} \text{ and } x \in V \\[0.5em] \text{(c) } a0 = 0 \ \ \forall a \in \mathbb{F}