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Row space of a matrix, column space

Row space of a matrix. The row space of a matrix is that subspace spanned by the rows of the matrix (rows viewed as vectors). It is that space defined by all linear combinations of the rows of the matrix.

Consider a matrix containing five rows and three columns. The rows may be viewed as 3-vectors spanning some subspace of three-dimensional space. If the rows contain three linearly independent vectors they span all of three-dimensional space. If the rows contain only two linearly independent vectors they span the subspace of three-dimensional space defined by these two vectors (some plane passing through the origin). If all rows are multiples of some one row they represent one-dimensional subspace of three dimensional space corresponding to some line passing through the origin.

The effect of the elementary row operations on a matrix is to produce other sets of rows in the
same row space. If the rows of a matrix A span some subspace K of n-space V_{n} then the
elementary row operations will produce another matrix whose row vectors span the same
subspace of V_{n} . Row-equivalent matrices have the same row space. The dimension of the row
space corresponds to the number of linearly independent vectors required to span the row space
— which is equal to the rank of the matrix.

Column space of a matrix. The column space of a matrix is the subspace spanned by the columns of the matrix (columns viewed as vectors).

Theorem. For any mxn matrix the dimension of its row space is equal to the dimension of its column space and both dimensions are equal to its rank. In other words the number of linearly independent rows in a matrix is equal to the number of linearly independent columns.

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