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The '''column space''' of an ''m''-by-''n''
matrix with
real entries is the
subspace of '''R'''
''m'' 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 '''R'''
''n'' to '''R'''
''m'', 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 = ['''a'''
1, ...., '''a'''
n], then Col Z = Span {'''a'''
1, ...., '''a'''
n}
See also
row space.
{| style="margin:0 auto;" align=center width=80% id=toc
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'''Topics in
mathematics related to
linear algebra'''
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Edit
|-
|align=center|
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
|}
