Find basis for row space using non-zero rows from RREF
Enter 2x2 Matrix
Enter 3x3 Matrix
Enter 3x2 Matrix
Result
Step-by-Step Derivation
Row Space Formula
Row(A) = span{ r₁, r₂, ..., rₘ }
dim(Row(A)) = rank(A) = number of non-zero rows in RREF
Basis = non-zero rows from RREF(A)
Row(A) = Column(A^T)
The row space of matrix A is the span of its row vectors. A basis is formed by the non-zero rows from the row-reduced echelon form (RREF) of A.
⚠Row space basis vectors come from RREF rows (not original matrix). This differs from column space where basis comes from original columns.
What is Row Space?
The row space of an m×n matrix A is the set of all possible linear combinations of its row vectors. Geometrically, it's a subspace of ℝⁿ spanned by the rows of A.
Non-zero Rows
Rows in RREF that contain leading 1s. These form a basis for row space.
Row Rank
Dimension of row space = rank(A) = number of non-zero rows in RREF.
Full Row Rank
When rank(A) = m (number of rows), all rows are linearly independent.
Transpose
Row space of A equals column space of A^T. Row rank = column rank.
Teaching Example: A = [[1,2],[2,4],[3,6]]
1. RREF(A) = [[1,2],[0,0],[0,0]]
2. Non-zero rows in RREF: [1, 2]
3. Row space basis = [1, 2]
4. Row rank = 1
Applications
Linear SystemsRank DeterminationData CompressionSignal ProcessingMachine Learning
Frequently Asked Questions
What is row space?▼
Row space of matrix A is the span of its row vectors. It contains all possible linear combinations of rows.
How to find row space basis?▼
1. Compute RREF of A. 2. Take non-zero rows from RREF. 3. These rows form a basis.
Row space vs column space?▼
Row space is span of rows. Column space is span of columns. Both have same dimension (rank).
Row space of A vs A^T?▼
Row space of A = column space of A^T. Row rank = column rank = rank(A).
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.