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The column space col(A) is the span of the columns of A; it equals the image of the linear transformation x ↦ Ax. The row space row(A) is the span of the rows of A, equal to col(Aᵀ). Both are fundamental subspaces determined by A; rank(A) is the dimension of each.
From your prerequisite on span of vectors, you know that the span of a set of vectors is all linear combinations of those vectors — it is the smallest subspace containing them all. The column space of a matrix A, written col(A), applies this idea directly: take the columns of A as your set of vectors and form their span. If A is m × n, its columns are vectors in ℝᵐ, so col(A) is a subspace of ℝᵐ. From your subspaces prerequisite, you know this span is guaranteed to be a subspace: it contains the zero vector and is closed under addition and scalar multiplication.
The geometric interpretation is the most important thing to understand. Multiplying A by a vector x computes a linear combination of the columns of A, where the entries of x are the coefficients. So the set of all vectors Ax, as x ranges over all of ℝⁿ, is exactly col(A). This means col(A) is the image of the linear transformation T(x) = Ax — the set of all outputs the transformation can possibly produce. The equation Ax = b has a solution if and only if b is in col(A). Geometrically: b must lie in the space that A's columns can reach.
The row space row(A) is the span of the rows of A, viewed as vectors in ℝⁿ. Equivalently, it is the column space of Aᵀ. A crucial fact: row reduction preserves the row space but can change the column space. When you row-reduce A to its echelon form, the nonzero rows of the echelon form span the same row space as the original rows of A. The pivot columns of A (not the echelon form) form a basis for the column space. This asymmetry is a source of many student errors — do not take the pivot columns from the reduced form and declare them a basis for col(A); go back to the original matrix.
The rank of A is defined as the dimension of the column space, and it equals the dimension of the row space — these two numbers always agree, a non-obvious theorem. The rank counts the number of pivot columns (or pivot rows) after row reduction. Together with the rank-nullity theorem, which you will use extensively in least-squares approximation, rank organizes all four fundamental subspaces of a matrix: the column space, the row space, the null space, and the left null space. Understanding which space a given vector belongs to — and why — is the unifying question of linear algebra.