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Encyclopedia > Column space

The column space of an m-by-n matrix with real entries is the subspace of Rm generated by the column vectors of the matrix. Its dimension is the rank of the matrix and is at most min(m,n).

If one considers the matrix as a linear transformation from Rn to Rm, then the column space of the matrix equals the image of this linear transformation.

The column spaces of a matrix Z is the set of all linear combinations of the columns in Z. If Z = [a1, ...., an], then Col Z = Span {a1, ...., an}

See also row space.

Topics in mathematics related to linear algebra

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Vectors | Vector spaces | Linear span | Linear transformation | Linear independence | Linear combination | Basis | Column space | Row space | Dual space | Orthogonality | Eigenvector | Eigenvalue | Least squares regressions | Outer product | Cross product | Dot product | Transpose | Matrix decomposition

  Results from FactBites:
NationMaster - Encyclopedia: Rank (matrix theory) (765 words)
In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent.
The maximal number of linearly independent columns of the m-by-n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of F
The column rank of a matrix A is the maximal number of linearly independent columns of A. The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix (this is the "rank theorem" or the "rank-nullity theorem").
Gas chromatography oven heaters - Patent 6485543 (3359 words)
The oven space is typically divided by a baffle 5 into a heater space 3 containing a heater element 6 and a fan 2, and a column space 4 containing one or more separation columns (not shown).
The baffle is also designed to permit flow from the heater space to the column space between the baffle periphery and the walls of the oven to permit a circulating flow of forced hot gas, typically air.
Positioning heater 60 in column space 4 such that heat is directed into the air stream directed toward space 3 promotes even heating because fan 2 mixes the heat energy with the heat energy generated by the heater in compartment 3 and directs the mixed heat energy to the column space 4.
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