Injective surjective

To remember these easily, use the "Location & Letter" method.

Think of the word "IS" (Injective/Surjective). It follows the order of an equation: $x \to b$ (Inputs to Outputs).

The Keyword: INjective = INput uniqueness.
The Location: Inputs are the Columns ( $x$ ).
The Pivot Trick: Pivot in every Column.
The Logic: If every column has a pivot, there are no "free variables." If there are no free variables, you can't have two different inputs giving the same output.
Shape: Usually Tall. You need enough "room" (rows) to give every input its own unique output.

The Keyword: SURjective = SURface (you cover the whole output surface).
The Location: Outputs are the Rows ( $b$ ).
The Pivot Trick: Pivot in every Row.
The Logic: If every row has a pivot, you can solve $A x = b$ for any $b$ . There are no "zero rows" at the bottom that make certain outputs impossible to reach.
Shape: Usually Wide. You need enough "power" (columns) to reach every dimension of the target space.

Word	Focus	Pivot Location	Visual
Injective	Inputs	Every Column	Tall (or square)
Surjective	Span (Rows)	Every Row	Wide (or square)

Injective (One-to-One): You fire arrows (inputs) at a target. It is injective if no two arrows hit the same spot. This depends on your aim (the columns).
Surjective (Onto): It is surjective if every inch of the target has at least one arrow in it. This depends on how big your quiver is compared to the target area (the rows).

The Pivot Rule:

Pivot in every Column? Each arrow has its own unique landing spot (Injective).
Pivot in every Row? No part of the target is left untouched (Surjective).