## null space

Let $\displaystyle A$ be a flat mXn matrix (m<n).

Let the QR decomposition of $\displaystyle A^{T}$ be

$\displaystyle A^{T} = \left \begin{bmatrix} Q_{1}&Q_{2} \end{bmatrix} \right \left \begin{bmatrix} R_{1}\\0 \end{bmatrix} \right$

And let the singular-value decomposition of $\displaystyle R_{1}$ be

$\displaystyle R_{1} = \left \begin{bmatrix} U_{1}&U_{2} \end{bmatrix} \right \left \begin{bmatrix} \Sigma_{1}&0 \\ 0&0 \end{bmatrix} \right \left \begin{bmatrix} V_{1}&V_{2} \end{bmatrix} \right^{T}$

Can you use the singular-value decomposition of $\displaystyle R_{1}$ to find the null space of $\displaystyle A$ ?

There's a bunch of things you already know from the decompositions:

$\displaystyle A^{T} = Q_{1}R_{1}$

$\displaystyle R_{1} = U_{1}\Sigma_{1}V_{1}$

$\displaystyle A^{T} = Q1_{1}U_{1}\Sigma_{1}V_{1}$

$\displaystyle AQ_{2} = 0$

$\displaystyle R_{1}V_{2} = 0$

$\displaystyle R_{1}^{T}U_{2} = 0$

$\displaystyle \text{null} (A^{T}) = \text{null} (R_{1})$

But I can't seem to combine the above in any meaningful way.