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Thread: $W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$, where $W_i$ ($V_i$) is

  1. #1
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    $W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$, where $W_i$ ($V_i$) is

    $W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$, where $W_i$ ($V_i$) is the space of row (column) vectors.


    How to show?


    Precisely, if $A_1,A_2$ are two $m\times n$ matrices. Let $W_1,W_2$ be the space of row vectors of $A_1$ and $A_2$ respectively; $V_1,V_2$ be the space of column vectors of $A_1$ and $A_2$ respectively. Show that $W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$.


    It sounds like $rank\left(A\atop B\right)=rank(A)+rank(B)$ if and only if $W_1\cap W_2=\{0\}$. But how to proceed?
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  2. #2
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    Re: $W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$, where $W_i$ ($V_i$) is

    $\displaystyle A_1= \begin{pmatrix}1 & 2 \\ 2 & 4\end{pmatrix} $

    $\displaystyle A_2= \begin{pmatrix}1 & 3 \\ 2 & 6\end{pmatrix} $

    $\displaystyle W_1\cap W_2=(0)$

    $\displaystyle V_1\cap V_2\neq (0)$
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