The row space of an m×n matrix A is the span of its row vectors, forming a subspace of Rⁿ. Unlike the column space, the row space IS preserved by elementary row operations, so a basis for the row space can be read directly as the nonzero rows of any row echelon form of A. The dimension of the row space equals the rank of A, confirming that row rank equals column rank — a foundational theorem. The row space and null space of A are orthogonal complements in Rⁿ: every vector in Rⁿ can be uniquely written as the sum of a row-space component and a null-space component.
Row-reduce A to REF and identify the nonzero rows as a row-space basis. Then verify the rank matches the column rank found from Col(A). Use the orthogonality relationship Nul(A) ⊥ Row(A) to geometrically motivate least squares.
You already know that the column space Col(A) is the span of the columns of A — the set of all vectors Ax as x ranges over Rⁿ. The row space Row(A) is the analogous object for rows: the span of the row vectors of A, which lives in Rⁿ since each row has n entries. Where the column space asks "which vectors in Rᵐ can A produce?", the row space asks "what linear combinations of the rows of A form this subspace of Rⁿ?" Both are subspaces associated with A, but they live in different ambient spaces and are found by different procedures.
The critical asymmetry is one you need to internalize carefully. From your work with column space, you know that elementary row operations change the column space — you must find pivot columns in the original A, not in the reduced form. The row space is the opposite: row operations preserve the row space. This is because each row operation (swap two rows, scale a row, add a multiple of one row to another) produces rows that are still linear combinations of the original rows — no new row directions are created or destroyed. So after row reducing A to echelon form, the nonzero rows directly give a basis for the row space. The rule is: row-space basis from the reduced form, column-space basis from the original.
There is a deep geometric relationship between the row space and the null space. Every vector x in Nul(A) satisfies Ax = 0, which means every row of A is orthogonal to x — each row dotted with x equals zero. Consequently, Nul(A) ⊥ Row(A): the null space and row space are orthogonal complements in Rⁿ. Their dimensions add to n (by the rank-nullity theorem), and every vector in Rⁿ decomposes uniquely into a row-space component and a null-space component. This decomposition is the foundation for least squares: the "best" solution to Ax = b lives in the row space of A, and the error is in the null space.
Finally, consider what these two subspaces reveal about rank. The number of nonzero rows in the echelon form counts the dimension of Row(A). The number of pivot columns counts the dimension of Col(A). These are the same number — the rank of A. This means row rank equals column rank, despite the two spaces living in different ambient spaces (Rⁿ and Rᵐ respectively). Both count the same thing: the number of truly independent constraints or directions in A. The equality of row and column rank is one of the more surprising theorems of linear algebra, and the row space is essential for understanding why it holds.
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