Shape and Rank Visualized as a Summary Table

A compact map of how matrix shape and rank control the main linear algebra cases

Matrix shape tells us the dimensions of the domain and codomain. Rank tells us how many independent directions survive the transformation. Together, shape and rank determine the dimensions of row space, column space, null space and left null space.

The summary table compares the main square, tall and short matrix cases side by side. It shows how the same rank facts control one-to-one behavior, onto behavior, solution structure, determinant, inverse and pseudoinverse formulas.

This visualization summarizes the main matrix shape-and-rank cases in one table, so the consequences of full rank, rank deficiency, tall shape and short shape can be compared directly.

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Summary table

The table organizes the six core cases: square full-rank, square rank-deficient, tall full-column-rank, tall rank-deficient, short full-row-rank and short rank-deficient matrices.

Summary table comparing matrix shape and rank cases, including null space, left null space, one-to-one, onto, solvability, determinant, inverse and pseudoinverse behavior.
Shape and rank summary table comparing domain decomposition, codomain decomposition, reduced echelon form, solvability, determinant, inverse and pseudoinverse behavior.

Concept

For an m × n matrix M of rank r, the null space has dimension n − r. This is the number of directions in the domain that collapse to zero. If n − r = 0, the transformation is one-to-one.

The left null space has dimension m − r. This is the number of codomain directions that are not reached by M. If m − r = 0, the transformation is onto.

Main cases

Square full-rank matrices are both one-to-one and onto, so they have nonzero determinant and an inverse. Tall full-column-rank matrices are one-to-one but not onto. Short full-row-rank matrices are onto but not one-to-one.

Rank-deficient matrices lose at least one independent direction. That loss appears as free variables in the domain, unreachable directions in the codomain, or both.

Key equations

dim null(M) = n − r
dim row(M) = r
dim col(M) = r
dim ℓ-null(M) = m − r
one-to-one ⇔ null(M) = {0} ⇔ r = n
onto ⇔ ℓ-null(M) = {0} ⇔ r = m

These equations explain why a single table can summarize so many consequences of shape and rank.

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References

Related chapter: GraphMath — Shape & rank