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Question

How to show that column span of matrices $B$ and $BB^{'}$ are same, where $B$ is a $m \times n$ matrix and $B^{'}$ is the transpose of $B$

Thoughts

I can see that the columns of $BB^{'}$ lie in the span of columns of $B$, hence span of columns of $BB^{'}$ lies inside span of columns of $B$. But how to see the equality

Debu
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1 Answers1

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Use ranks, the ranks of $B$ and $BB^\intercal$ are the same; also, the column span is the image vector space, so you will have that $\mathsf{im}(BB^\intercal) \subset \mathsf{im}(B)$ and that the two vector spaces have the same dimensions.

To prove that $\mathsf{rk}(BB^\intercal) = \mathsf{rk}(B),$ consider $x$ such that $BB^\intercal x = 0,$ then $x^\intercal BB^\intercal x = 0,$ or $\mathsf{norm}(B^\intercal x)^2 = 0,$ so $B^\intercal x = 0$ as well. Therefore $\mathsf{ker}(BB^\intercal) \subset \mathsf{ker}(B^\intercal).$ Therefore, the rank of $BB^\intercal$ is $\leq \mathsf{rk}(B)$ and the nullity of $BB^\intercal$ is also $\leq \mathsf{nul}(B^\intercal) = \mathsf{nul}(B),$ so they each must coincide with the corresponding number, given that $\mathsf{rk}+\mathsf{nul}$ must equal the dimension of the vector space.

William M.
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  • But this proof works only when $B$ is square right? – Debu May 06 '24 at 15:54
  • I am thinking perhaps the question implies that $B$ is square. – Debu May 06 '24 at 15:55
  • Rank + nullity equal to dimension is a general result. – William M. May 06 '24 at 15:56
  • yeah, but see $B^{'}$ acts on $x$ and also $B$ acts on $x$ in the proof you have formed above. So that implies that the shape of $B^{'}$ and $B$ is same, right? – Debu May 06 '24 at 15:58
  • @Debu I noticed my mistake. Check again. – William M. May 06 '24 at 15:58
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    $null(BB^{'}) \leq null(B^{'})$ $\implies rank(BB^{'}) \geq rank(B^{'}) \implies rank(BB^{'}) \geq rank(B)$ but we had $dim \ col (BB^{'}) \leq dim \ col(B) \implies dim \ col(BB^{'}) = dim \ col(B)$ but as span of column of $BB^{'}$ lies inside span of column of $B$, they are same vector space. – Debu May 06 '24 at 16:09