A sum U plus W is direct exactly when every vector has at most one decomposition. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.
The data
A linear map \(T:V\to W\) is organized by its kernel \(\ker T\) and image \(\operatorname{im}T\). Quotients and duals express the same information without choosing bases.
A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.
Derivation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Invariant content
The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.
Scope
Dimension formulas below assume finite-dimensional spaces. Infinite-dimensional vector spaces require cardinal arithmetic and may not identify naturally with their double duals.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.