Column Space
Calculate the column space, rank, and determinant of any 2x2 matrix instantly. Free online linear algebra tool for students studying vector spaces and matrix theory.
About This Calculator
The column space calculator computes the fundamental subspaces of a 2x2 matrix. Enter the four entries of your matrix and instantly get the column space basis, the rank, and the determinant. This tool is designed for linear algebra students, engineers, data scientists, and anyone working with vector spaces.
The column space (or range) of a matrix A is the span of its column vectors. For a 2x2 matrix [[a11, a12], [a21, a22]], the column space is the set of all vectors that can be written as a linear combination of col1 = (a11, a21) and col2 = (a12, a22). The calculator determines whether the two columns are linearly independent (full rank 2) or dependent (rank 1). If the determinant is nonzero, both columns form a basis. If the determinant is zero, the calculator selects the first nonzero column as the basis vector.
The rank of the matrix -- the dimension of the column space -- is displayed alongside the determinant, which indicates whether the matrix is invertible. A nonzero determinant means the matrix has full rank and an inverse exists.
Frequently Asked Questions
What is the column space of a matrix?
The column space of a matrix is the set of all possible linear combinations of its column vectors. For a 2x2 matrix [[a,b],[c,d]], the column space is the span of the two column vectors col1=(a,c) and col2=(b,d). Its dimension equals the rank of the matrix.
How do you find the basis for the column space?
To find a basis for the column space of a 2x2 matrix, compute the determinant. If the determinant is nonzero, both columns are linearly independent and form the basis. If the determinant is zero, the columns are dependent; pick the first non-zero column as the basis vector. If the matrix is all zeros, the column space is 0.
What is the relationship between rank and column space?
The rank of a matrix equals the dimension of its column space. For a 2x2 matrix, the rank is 2 when both columns are linearly independent (nonzero determinant), 1 when only one column is independent (zero determinant but at least one nonzero entry), and 0 for the zero matrix.
Is this column space calculator free to use?
Yes, this column space calculator is completely free to use with no registration or login required. You can compute as many matrices as you need.
What does a zero determinant tell me about the column space?
A zero determinant means the columns are linearly dependent -- one column is a scalar multiple of the other. In this case the column space has dimension 1 (rank 1), and you only need one basis vector to span the entire column space.
Can I share my column space calculation results?
Yes, you can share your results by copying the page URL after calculation. The matrix values are saved in the URL, so anyone with the link will see the same computation.