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Define the column space of a matrix.

The column space of a matrix is the span of its column vectors.

The column space of a matrix A, denoted by Col(A), is the set of all linear combinations of its column vectors. In other words, Col(A) is the span of the column vectors of A. Geometrically, the column space of a matrix represents the subspace of the vector space that is spanned by the columns of the matrix.

To find the column space of a matrix, we can perform row operations on the matrix to obtain its reduced row echelon form (RREF). The non-zero rows of the RREF correspond to linearly independent columns of the original matrix, and the column space is the span of these columns.

Alternatively, we can find a basis for the column space by finding the pivot columns of the RREF. The pivot columns are the columns that contain a leading non-zero entry in the RREF. The column space is then the span of these pivot columns.

The column space of a matrix is an important concept in linear algebra, as it provides information about the range of the linear transformation represented by the matrix. In particular, the dimension of the column space is equal to the rank of the matrix, which is a fundamental property of linear transformations.

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