(a) The row space of a matrix is isomorphic to the column space of its transpose.
(b) The row space of a matrix is isomorphic to its column space.
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We have the following general theorems that are usually proved in any Linear Algebra course:
1. Two finite-dimensiona real vector spaces V and W are isomorphic if amd only if they have the same dimension, that is,
dim V = dim W = n. As a Corollary, every n-dimensional real vector space is isomorphic to R^n.
2. The column rank of any (m by n) matrix A, that is, the maxinmal number of linearly independent columns of A, is the same as the row-rank of A, that is, the maximal number of linearly independent rows of A.
Now, the row space of a matrix is the set of all linear combinations of its rows, considered elements of R^m. Suppose, A has row-rank k, that is, A has k linearly ...
General theorems for linear algebra are carefully reviewed in the solution.