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I've read a text, and they say the following lemma,

Lemma: For given matrices $A,B$ , the matrix $$ \begin{pmatrix} A & B\\ 0^T & v^T\\ \end{pmatrix} $$ has full row rank for all nonzero $v$ iif $A$ has full row rank.

where the matrix, not needed to be squared. can give me som hint, thanks!

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  • $\begingroup$ Hint: A matrix $M$ has full row-rank if and only if the system $M^Tx = 0$ (or equivalently $xM = 0$ for a row-vector $x$) has only the trivial solution $\endgroup$ May 28 '21 at 19:47
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If $A$ does not have full row rank, then there exists a nonzero $u$ such that $u^\top A = 0$. Try to come up with a nonzero vector $v$ and scalar $c$ such that $$\begin{pmatrix} u^\top & c \end{pmatrix} \begin{pmatrix} A & B \\ 0^\top & v^\top \end{pmatrix} = 0.$$ (It will depend on what $u^\top B$ is.)


If there exists nonzero $v$ such that $M := \begin{pmatrix} A & B \\ 0^\top & v^\top \end{pmatrix}$ does not have full row rank, then there exists a vector $u$ and scalar $c$ such that $\begin{pmatrix}u \\ c \end{pmatrix}$ is nonzero and $$\begin{pmatrix} u^\top & c \end{pmatrix} \begin{pmatrix} A & B \\ 0^\top & v^\top \end{pmatrix} = 0.$$

Show that the above conditions imply that $u$ is nonzero and that $u^\top A = 0$.

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call your overall matrix $C$ which has $m$ rows .

1.) $A \text{ is full row rank }\implies C \text{ is full row rank}$
since $A$ has full row rank, it is surjective and has a right inverse. Thus there is some (block) matrix $W$ such that
$CW = \begin{pmatrix} A & B\\ \mathbf 0^T & \mathbf v^T\\ \end{pmatrix}W = \begin{pmatrix} I_{m-1} & *\\ \mathbf 0^T & \mathbf v^T\\ \end{pmatrix}$

$m = \text{rank}\big(CW\big)\leq \text{rank}\big(C\big)\leq m$

2.) $C \text{ is full row rank} \implies A\text{ is full row rank } $
$C$ is sujective so it has a right inverse $W'$
$CW' = I_m$
This implies that for all non-zero $\mathbf x':=\begin{pmatrix}\mathbf x \\ 0 \end{pmatrix}$
$(\mathbf x')^TC \neq\mathbf 0^T\implies \mathbf x^TA\neq \mathbf 0^T$
and hence all rows of $A$ are linearly independent.

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