๐ Column Space Calculator
Find the column space, basis, and rank of a matrix
How to Use This Calculator
Select Matrix Dimensions
Choose the number of rows and columns for your matrix.
Enter Matrix Elements
Input all elements of your matrix.
Calculate
Click to compute the column space, basis vectors, rank, and RREF.
Review Results
The pivot columns form a basis for the column space. Rank = number of pivot columns.
Formula
Column Space = Span{column vectors of A}
Rank = Dimension of Column Space = Number of Pivot Columns
Definition:
The column space (range) of a matrix A is the span of its column vectors. It's the set of all linear combinations of columns.
Finding Basis:
1. Perform Gaussian elimination to get RREF
2. Identify pivot columns
3. The original columns corresponding to pivot positions form a basis
Properties:
- Rank(A) = dim(Column Space) = dim(Row Space)
- Column Space = {Ax : x โ โโฟ}
- Dimension = number of linearly independent columns
About Column Space Calculator
The Column Space Calculator finds the column space (also called range) of a matrix. The column space is the span of the column vectors - the set of all possible linear combinations. It's a fundamental subspace in linear algebra, equal in dimension to the rank of the matrix.
When to Use This Calculator
- Linear Algebra: Find basis for column space
- Systems of Equations: Determine if system is solvable
- Rank Analysis: Find matrix rank
- Vector Spaces: Understand fundamental subspaces
- Image/Range: Find the range of a linear transformation
Why Use Our Calculator?
- โ Complete Analysis: Shows basis, rank, dimension, and RREF
- โ Pivot Columns: Identifies linearly independent columns
- โ Visual Display: Clear presentation of results
- โ Educational: Helps understand column space concept
- โ Accurate: Precise Gaussian elimination
- โ Free: No registration required
Key Concepts
- Column Space: Span of column vectors, denoted Col(A) or Range(A)
- Basis: Linearly independent columns corresponding to pivot positions
- Rank: Dimension of column space = number of pivot columns
- RREF: Reduced row echelon form reveals pivot structure
- Fundamental Theorem: Rank(A) = dim(Col(A)) = dim(Row(A))
Applications
System Solvability: Ax = b has a solution if and only if b โ Col(A).
Linear Transformations: Column space is the image/range of the transformation T(x) = Ax.
Frequently Asked Questions
What is the column space?
The column space of a matrix A is the span of its column vectors - all possible linear combinations of columns. It's the set {Ax : x โ โโฟ}.
How do I find the column space?
Perform Gaussian elimination to RREF, identify pivot columns, then the original columns at those positions form a basis for the column space.
What's the relationship between rank and column space?
Rank = dimension of column space = number of linearly independent columns = number of pivot columns in RREF.
Is column space the same as row space?
No, but they have the same dimension (rank). Column space is a subspace of โแต (where m is number of rows), while row space is a subspace of โโฟ (where n is number of columns).
Can I use pivot columns directly from RREF as basis?
No! Use the original columns (before elimination) at the pivot positions. RREF columns may not be in the original column space.