Meaning of row space
I used to think of the row space of an $m \times n$ matrix $A$ as the column space of $A^T$, and therefore the row vectors are the images of the standard dual basis of $\mathbb{R}^m$ under $A^T$.
But it seems that we can have an interpretation of the row space without introducing the dual space.
I think, but am not sure, that the row spaces are those vectors which are 1-1 mapped to vectors in the column space.
Is this correct?
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$\begingroup$The column space of a matrix $ A $ is the set of all vectors $ y $ such there $ y $ is a linear combination of the columns of $ A $. In other words, it is the set of all vectors $ y $ such that there is a vector $ x $ with $ A x = y $.
What does that mean? It means that it is the set of all vector for which $ A x = y $ has a solution. If you ask the question "for what vectors $ y $ does $ A x = y $ have a solution?" the answer is "for all vectors in the columns space".
If you think of the rows of matrix $ A $ as vectors, then the row space is the set of all vectors that are linear combinations of the rows. In other words, it is the set of all vectors $ y $ such that $ A^T x = y $ for some vector $ x $.
How are they related? You can prove that if $ y $ is in the column space of $ A $, then any solution to $ A x = y $ can be written as $ x = x_r + x_n $, where $ x_r $ is in the row space of $ A $ and $ x_n $ is in the null space ($ A x_n = 0 $).
$\endgroup$ $\begingroup$The row space of an m-by-n matrix is the linear subspace generated by row vectors of the matrix.
that the row spaces are those vectors which are 1-1 mapped to vectors in the column space.
I cannot make sense of this statement. However,
one has the theorem that the row space of $A$ and the column space of $A$ are of the same dimension. Also, note that if two vector spaces are of the same finite dimension, they are isomorphic.
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