Find basis for column space (range) using pivot columns from RREF
Enter 2x2 Matrix
Enter 3x3 Matrix
Enter 3x2 Matrix
Result
Step-by-Step Derivation
Column Space Formula
C(A) = span{ a₁, a₂, ..., aₙ }
dim(C(A)) = rank(A) = number of pivot cols
Basis = pivot columns from original A
C(A) = { Ax | x ∈ ℝⁿ }
The column space (range) of matrix A is the span of its column vectors. A basis is formed by the pivot columns from the original matrix after computing RREF.
⚠Column space basis vectors come from the ORIGINAL matrix columns, not from RREF. Only column positions matter from RREF.
What is Column Space?
The column space (or range) of an m×n matrix A is the set of all possible linear combinations of its column vectors. Geometrically, it's a subspace of ℝᵐ spanned by the columns of A.
Pivot Columns
Columns with leading 1s in RREF. These form a basis for column space.
Column Rank
Dimension of column space = rank(A) = number of pivot columns.
Full Column Rank
When rank(A) = n (number of columns), all columns are linearly independent.
Range
Column space equals the range of the linear transformation defined by A.
Teaching Example: A = [[1,2],[2,4],[3,6]]
1. RREF(A) = [[1,2],[0,0],[0,0]]
2. Pivot column: 1 only
3. Column space basis = first column of A = [1, 2, 3]^T
4. Column rank = 1
Applications
Linear SystemsLeast SquaresData AnalysisComputer GraphicsMachine Learning
Frequently Asked Questions
What is column space?▼
Column space (range) of matrix A is the span of its column vectors. It contains all possible linear combinations of columns.
How to find column space basis?▼
1. Compute RREF of A. 2. Identify pivot columns in original matrix. 3. Those columns form a basis.
Column space vs row space?▼
Column space is span of columns. Row space is span of rows. They have same dimension (rank).
What is column rank?▼
Column rank = dimension of column space = number of pivot columns = rank(A).
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