In linear algebra, the column space of a matrix (sometimes called the range of a matrix) is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of mdimensional Euclidean space. The dimension of the column space is called the rank of the matrix.^{[1]}
The column space of a matrix is the image or range of the corresponding matrix transformation.
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Definition
Let A be an m × n matrix, with column vectors v_{1}, v_{2}, ..., v_{n}. A linear combination of these vectors is any vector of the form
where c_{1}, c_{2}, ..., c_{n} are scalars. The set of all possible linear combinations of v_{1},...,v_{n} is called the column space of A. That is, the column space of A is the span of the vectors v_{1},...,v_{n}.
Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:
Therefore, the column space of A consists of all possible products Ax, for x ∈ R^{n}. This is the same as the image (or range) of the corresponding matrix transformation.
Basis
The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space.
For example, consider the matrix
The columns of this matrix span the column space, but they may not be linearly independent, in which case some subset of them will form a basis. To find this basis, we reduce A to reduced row echelon form:
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