Shape and rank already determine a large part of the story. They control the dimensions of row space, column space, null space and left null space, and from those dimensions follow one-to-one, onto, solvability, determinant and inverse behavior. This chapter organizes the main cases side by side.
Matrix shape tells us the sizes of the domain and codomain, while rank tells us how many independent directions survive the transformation.
The chapter compares these facts across the core shape-and-rank cases rather than treating each property in isolation.
Because once the dimensions of the four fundamental subspaces are fixed, many consequences become unavoidable. Free variables, unreachable outputs, uniqueness of solutions and invertibility all follow from how much dimension is lost and where that loss occurs.
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Full column rank gives one-to-one, because null space is trivial. Full row rank gives onto, because left null space is trivial. Only in the square full-rank case do both happen at once.