I am trying to understand the row space of a matrix. More specifically what does it mean? I mean what is its significance. For example, the column space of a matrix denotes space spanned by the columns of the matrix. And how null space is related to row space as there is a theorem about them which states, 'Space which is perpendicular to row space of A is equivalent to null space of A"
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In fact it's pretty clear from the definitions that the orthogonal complement ("space perpendicular to") of the row space is equal to the nullspace, not just "equivalent to". – David C. Ullrich Mar 17 '20 at 12:24
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Sir, how it's so clear from the definition? I mean it's very hard to think of the row space and then relating it to null space – Merajul Arefin Pial Mar 18 '20 at 13:47
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It's clear from the definition_s_, including the definition of matrix multiplication. See new answer... – David C. Ullrich Mar 19 '20 at 10:38
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If the column space is "the space spanned by the columns of the matrix", then the row space is the space spanned by the rows of the matrix. Regarding your second statement, there is indeed a theorem; see the "fundamental theorem" of linear algebra.
Ben Grossmann
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Sir, I know this theorem, but its not very understandable for me. How the row space of a matrix become perpendicular with the null space of that matrix? – Merajul Arefin Pial Mar 17 '20 at 11:28
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@MerajulArefinPial See this post. Does that answer your question? – Ben Grossmann Mar 17 '20 at 11:34
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Say $R$ and $N$ are the row space and null space of $A$. It really is more or less obvious that $N=R^\perp$:
If the rows of $A$ are $R_1,\dots,R_n$ and $x$ is a column vector then $$Ax=\begin{pmatrix}R_1\cdot x\\ \vdots \\R_n\cdot x\end{pmatrix}.$$So $x\in N$ (ie $Ax=0$) if and only if $R_j\cdot x=0$ for every $j$, which says exactly $x\in R^\perp$.
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David C. Ullrich
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