seniorLinear Algebra
What is the difference between row space and column space of a matrix?
Updated May 16, 2026
Short answer
Column space is all outputs Ax can produce, row space is span of row vectors representing constraints.
Deep explanation
The column space of a matrix A defines all possible outputs of the linear transformation Ax, while the row space represents the space spanned by row vectors and encodes constraints on solutions. Both have the same dimension equal to rank(A), but live in different vector spaces. Column space lives in output space, row space lives in input constraint space.
Real-world example
Used in solving Ax=b where column space determines if solution exists.
Common mistakes
- Thinking row space and column space are identical sets.
Follow-up questions
- Why do row rank and column rank always match?
- Which space matters for solving Ax=b?