Enter Matrix Data
Use spaces, commas, or semicolons between entries. Fractions like 1/2 and -3/4 are supported.
Example Data Table
| Example | Matrix | Expected Rank | Possible Row Space Basis |
|---|---|---|---|
| Example 1 | [1 2 3; 2 4 6; 1 1 0] |
2 | [1 0 -3], [0 1 3] |
| Example 2 | [1 0 2 4; 0 1 3 5; 0 0 0 0] |
2 | [1 0 2 4], [0 1 3 5] |
| Example 3 | [2 4; 1 2] |
1 | [1 2] |
Formula Used
The calculator uses row-reduction because elementary row operations preserve the row space of a matrix.
Core idea: Row(A) = span{nonzero rows of RREF(A)}
Rank: rank(A) = number of nonzero rows in RREF(A)
Nullity: nullity(A) = number of columns - rank(A)
After converting the matrix to reduced row echelon form, every nonzero row becomes a basis vector for the row space.
How to Use This Calculator
- Enter the matrix dimensions in the rows and columns boxes.
- Type your matrix in the text area, one row per line.
- Separate values with spaces, commas, or semicolons.
- Choose decimal or fraction-style output.
- Adjust precision and tolerance if needed.
- Click Calculate Row Space to see the rank, pivot columns, RREF, and basis rows above the form.
- Use the CSV or PDF buttons to download the calculated result.
FAQs
1. What is the row space of a matrix?
The row space is the set of all linear combinations of the matrix rows. It describes every vector that can be formed by combining those rows.
2. Why is RREF useful for row space?
RREF makes independent rows easy to identify. The nonzero rows of the reduced matrix form a clean basis for the same row space.
3. Does changing row order affect row space?
No. Swapping rows only changes presentation. The span of the rows remains the same, so the row space does not change.
4. Are zero rows included in the basis?
No. Zero rows add no new direction to the span. Only nonzero independent rows are kept in a row space basis.
5. Is rank the same as row space dimension?
Yes. The rank equals the number of independent rows, which is exactly the dimension of the row space.
6. Can I enter fractions instead of decimals?
Yes. Entries such as 1/2, -3/5, and 7/4 are accepted. They are converted internally before row reduction starts.
7. What happens if some rows are dependent?
Dependent rows reduce to zero rows or combinations of pivot rows during elimination. They do not appear in the final basis.
8. Why download CSV or PDF results?
Exports help with homework records, reports, teaching notes, and verification. They also make it easier to share the matrix and computed basis.