Column space

The column space of a matrix is the set of all linear combinations of its columns. It is a subspace containing exactly the vectors the matrix can produce.

Linear combination: A sum of vectors multiplied by scalar coefficients. Every vector in a column space is formed by combining columns this way.
Matrix-vector multiplication: The operation that maps a vector of coefficients to a linear combination of a matrix’s columns. Its possible outputs are exactly the vectors in the column space.
Fundamental theorem of linear algebra: A set of relationships connecting a matrix’s four fundamental subspaces and their dimensions. It places the column space alongside the row space and the two null spaces.
Row space: The span of a matrix’s rows, regarded as vectors in the input-coordinate space. It is another fundamental subspace, but its vectors live in a different coordinate space.
Span (linear algebra): The set of all linear combinations of a given set of vectors. The column space is the span of the matrix’s columns.
Rank (linear algebra): The dimension of a matrix’s column space, equal to the dimension of its row space. Rank counts the independent directions spanned by the columns.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. A matrix’s column space is the image of its associated linear transformation.
Left null space: The null space of the transpose of a matrix. It consists of vectors orthogonal to every vector in the column space.
Subspace: A subset of a vector space that is itself a vector space under the same operations. Column spaces satisfy the closure properties that define subspaces.
Gaussian elimination: A sequence of row operations that reduces a matrix to echelon form. Pivot columns identify a basis for the original matrix’s column space.

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