ROW AND COLUMN SPACES

(Redirected from Row space)
The 'row space' of an ''m''-by-''n'' matrix with real entries is the subspace of 'R'^''n'' generated by the row vectors of the matrix. Its dimension is equal to the rank of the matrix and is at most min(''m'',''n'').
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 space of a matrix A is the set of all linear combinations of the columns in A. If A = ['a'₁, ...., 'a'_n], then Col A = Span {'a'₁, ...., 'a'_n}
The concept of 'row space' generalises to matrices over any field, in particular 'C', the field of complex numbers.
Intuitively, given a matrix A, the action of the matrix 'A' on a vector 'x' will (1) first project 'x' into the 'row space' of A, (2) perform an invertible transformation, and (3) place the resulting vector 'y' in the 'column space' of 'A'. Thus the result ' y =A x' must reside in the 'column space' of A.

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Example

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Example

Given a matrix J:
:$$
J =
egin{bmatrix}
2 & 4 & 1 & 3 & 2\
-1 & -2 & 1 & 0 & 5\
1 & 6 & 2 & 2 & 2\
3 & 6 & 2 & 5 & 1
end{bmatrix}

the rows are
'r'₁ = (2,4,1,3,2),
'r'₂ = (−1,−2,1,0,5),
'r'₃ = (1,6,2,2,2),
'r'₄ = (3,6,2,5,1).
Consequently the row space of J is the subspace of 'R'⁵ spanned by { 'r'₁, 'r'₂, 'r'₃, 'r'₄ }.
Since these four row vectors are linearly independent, the row space is 4-dimensional. Moreover in this case it can be seen that they are all orthogonal to the vector 'n' = (6,−1,4,−4,0), so it can be deduced that the row space consists of all vectors in 'R'⁵ that are orthogonal to 'n'.
See also null space.

External links

★ MIT Video Lecture on Column Space and Nullspace at Google Video, from MIT OpenCourseWare