Column Space Calculator
Find column space, basis vectors, dimension, and rank with step-by-step solutions
Matrix Input & Configuration
Highlighted columns form the basis for the column space
Quick Examples
Column Space Analysis
Basis for Column Space
Analysis
Key Concepts
Column Space
The span of all column vectors - all possible linear combinations of the columns.
Basis
Minimal set of linearly independent vectors that span the column space.
Dimension
Number of vectors in any basis for the column space (equals rank).
Mathematical Formulas
Col(A) = span{v₁, v₂, ..., vₙ}
Column space definition
dim(Col(A)) = rank(A)
Dimension equals rank
w = α₁v₁ + α₂v₂ + ... + αₙvₙ
Linear combination
How to Find Column Space
Write matrix with column vectors
Apply Gaussian elimination to get RREF
Identify pivot columns (leading 1's)
Corresponding original columns form the basis
Count basis vectors to get dimension
Understanding Column Space
What is Column Space?
The column space of a matrix A (denoted Col(A)) is the set of all possible linear combinations of its column vectors. It's a subspace of Euclidean space that represents all vectors that can be "reached" by the matrix through matrix-vector multiplication.
Why is it Important?
- •Determines solvability of linear systems Ax = b
- •Represents the range/image of linear transformations
- •Essential for understanding matrix rank and dimension
- •Critical in applications like data analysis and machine learning
Finding the Basis
The basis for the column space is found using Gaussian elimination. The columns that contain leading 1's (pivots) in the reduced row echelon form correspond to the linearly independent columns in the original matrix.
Linear Independence
Vectors are linearly independent if no vector can be written as a combination of the others.
Span
The span is the set of all linear combinations of the given vectors.
Dimension
The dimension equals the number of vectors in any basis for the space.
Example: 3×3 Matrix
Original Matrix
[3 7 -1]
[-2 1 12]
Column vectors: (1,3,-2), (4,7,1), (3,-1,12)
After RREF
[0 1 *]
[0 0 0]
Pivots in columns 1 and 2 → Basis: first two original columns
Result: Column space has dimension 2, spanned by vectors (1,3,-2) and (4,7,1).