What is the range of a matrix?
In linear algebra, the range of a matrix is the set of all output vectors you can produce by multiplying that matrix by an input vector. If a matrix is called A, then the range is all vectors of the form Ax. This set is also called the column space of the matrix.
The most practical number associated with this idea is the dimension of the range, which is the rank. So if you are looking for the range of a matrix in a calculator, most often you need the matrix rank and (optionally) a basis for the column space.
How this range of a matrix calculator works
This calculator parses your matrix, performs Gaussian elimination to find the reduced row echelon form (RREF), and then:
- Computes the rank (dimension of the range).
- Finds the pivot columns.
- Reports a basis for the range using the original matrix columns at those pivot positions.
- Shows the nullity, using the rank-nullity relationship.
How to use the calculator
Input format
Enter one row per line. Separate numbers by spaces or commas. You can also separate rows with semicolons on one line. All rows must have the same number of entries.
Example input
1 2 3
2 4 6
1 1 1
For this matrix, the calculator will find rank 2, meaning the range is a 2-dimensional subspace.
Why rank equals the dimension of the range
The columns of a matrix span the range. Some columns can be dependent combinations of others, so not every column adds a new direction. The number of independent column directions is exactly the rank. That is why rank tells you the dimension of the range.
Interpreting your result
- Rank = 0: only possible for the zero matrix; range contains only the zero vector.
- Rank = number of columns: full column rank; all columns are independent.
- Rank = number of rows (for a wide matrix maybe not possible unless rows ≤ cols): full row rank.
- Nullity = columns − rank: number of free variables in solutions to Ax = 0.
Common mistakes to avoid
- Mixing row lengths (e.g., one row has 3 entries and another has 4).
- Using symbols instead of numbers (like a, b).
- Confusing the range with the null space.
- Using a tolerance that is too large for your data scale, which can hide real pivots.
Quick conceptual checklist
- Write the matrix clearly.
- Reduce to RREF (or equivalent elimination).
- Count pivot columns to get rank.
- Use original pivot columns as a basis for the range.
Final note
If you are solving systems, fitting models, or analyzing transformations, knowing the range gives you immediate insight into what outputs are possible. Use this calculator whenever you need a fast, reliable rank and column-space summary.