What Is Basis for the Range of a Matrix?
The range of a matrix, also called its column space, is the set of all linear combinations of its columns. A basis for it is a smallest set of columns that spans that space.
Enter your matrix and the calculator row-reduces it to RREF, finds the pivot columns and then takes the matching columns of the original matrix as the basis. The count of pivots is the rank, and the nullity is the number of columns minus the rank.
Remember to take the basis from the original columns, not from the reduced matrix. Row operations change the column space.
How to Use the Basis for the Range of a Matrix Calculator
- Enter the matrix.
- Read the RREF and the pivot columns.
- Take those columns of the original matrix as the basis.
- Check the rank and nullity.
Basis for the Range of a Matrix Formula
Rank–nullity theorem
Basis for the Range of a Matrix Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| A | The matrix | |
| rank(A) | Number of pivot columns | |
| nullity(A) | Columns minus rank | |
| RREF | Reduced row echelon form |
Basis for the Range of a Matrix Worked Example
For the rows (1 2 3 4), (2 4 7 8), (3 6 10 12) the RREF has pivots in columns 1 and 3, so the rank is 2. A basis for the range is the original columns 1 and 3: (1, 2, 3) and (3, 7, 10). The nullity is 4 − 2 = 2.
With the example values loaded in the calculator above, the steps are:
- Row-reduce the matrix to reduced row echelon form (RREF).
- The pivots are in columns 1, 3, so the rank is 2.
- The basis uses the original columns of the matrix that sit in the pivot positions (not the columns of the RREF).
- Basis: (1, 2, 3), (3, 7, 10).
Result: Rank = dimension of the column space = 2; Pivot columns = 1, 3; Basis for the range (column space) = (1, 2, 3) (3, 7, 10).
How to Read the Basis and Rank
The pivot columns of the reduced form tell you which columns of the original matrix are independent. For the matrix with rows (1 2 3 4), (2 4 7 8) and (3 6 10 12), the pivots are in columns 1 and 3, so the rank is 2. A basis for the range is the original columns 1 and 3.
Rank plus nullity always equals the number of columns. Here 2 + 2 = 4. If the rank equals the number of columns, the columns are independent and the matrix has full column rank.
Where Basis for the Range of a Matrix Is Used
Linear algebra homework
Check a basis, the rank and the nullity against the full RREF working.
Solvability of Ax = b
A system has a solution exactly when b lies in the range. The basis shows what vectors can be reached.
Spotting redundant columns
Columns that are not pivot columns are combinations of the earlier ones, so they add no new direction.
Dimension counting
Confirm that rank plus nullity equals the number of columns.
Common Basis for the Range of a Matrix Mistakes to Avoid
- Take the basis from the original columns, not from the reduced matrix. Row operations change the column space.
- The pivot columns are found from the RREF, but the basis vectors come from the matrix you started with.
- The range is a subspace of the output space. Its dimension is the rank, which cannot exceed the number of rows or columns.
- Enter one row per line with entries separated by spaces or commas.
Basis for the Range of a Matrix Terms and Definitions
- Range
- The set of all linear combinations of the columns, also called the column space.
Basis for the Range of a Matrix FAQs
How do you find a basis for the range of a matrix?
Row-reduce the matrix to RREF, find the pivot columns, and take the columns in those same positions from the original matrix. Those columns are independent and span the range.
What is the dimension of the range?
It equals the rank, which is the number of pivots in the RREF. For a 3 × 4 matrix the rank is at most 3.
Is the range the same as the column space?
Yes. The range of the map x → Ax is the set of all combinations of the columns of A, which is the column space.
What is the nullity?
It is the dimension of the null space, the number of columns minus the rank. By the rank-nullity theorem rank + nullity = number of columns.