Row and Column Spaces
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The row and column space of an m-by-n matrix with real entries is the subspace of Rn generated by the row vectors and column vectors, respectiviely, of the matrix. Its dimension is equal to the rank of the matrix and is at most min(m,n). Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it will (1) first proje...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The row and column space of an m-by-n matrix with real entries is the subspace of Rn generated by the row vectors and column vectors, respectiviely, of the matrix. Its dimension is equal to the rank of the matrix and is at most min(m,n). Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it 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. See the singular value decomposition for more details on this second interpretation.
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