How is the null space related to singular value decomposition? It is said that a matrix's null space can be derived from QR or SVD. I tried an example:
$$A= \begin{bmatrix} 
1&3\\
1&2\\
1&-1\\
2&1\\
\end{bmatrix} 
$$ 
I'm convinced that QR (more precisely, the last two columns of Q) gives the null space:
$$Q= \begin{bmatrix} 
-0.37796&   -0.68252&   -0.17643&   -0.60015\\   
-0.37796&   -0.36401&   0.73034&   0.43731\\   
-0.37796&   0.59152&   0.43629&   -0.56293\\   
-0.75593&   0.22751&   -0.4951&   0.36288\\   
\end{bmatrix} 
$$
However, neither $U$ nor $V$ produced by SVD ($A=U\Sigma V^*$) make $A$ zero (I tested with 3 libraries: JAMA, EJML, and Commons):
$$ U= \begin{bmatrix} 
0.73039&   0.27429\\   
0.52378&   0.03187\\   
-0.09603&   -0.69536\\   
0.42775&   -0.66349\\
\end{bmatrix} 
$$
$$ \Sigma= \begin{bmatrix} 
4.26745&   0\\   
0&   1.94651\\   
\end{bmatrix} 
$$
$$ V= \begin{bmatrix} 
0.47186&   -0.88167\\   
0.88167&   0.47186\\  
\end{bmatrix} 
$$
This is contradiction to

Using the SVD, if $A=U\Sigma V^*$, then columns of $V^*$ corresponding to small singular values (i.e., small diagonal entries of $\Sigma$
  ) make up the a basis for the null space.

 A: For an $m \times n$ matrix, where $m >= n$,
the "full" SVD is given by
$$
A = U\Sigma V^t
$$
where $U$ is an $m \times m$ matrix, $\Sigma$ is an
$m \times n$ matrix and $V$ is an $n \times n$ matrix.
You have calculated the "economical" version of the SVD
where $U$ is an $m \times n$ and $S$ is $n \times n$.
Thus, you have missed the information about the left null space
given by the "full" matrix $U$.
The full SVD is given by
$$
U =
\left[
\begin{array}{cc}
   -0.7304 &  -0.2743 &  -0.1764 &  -0.6001 \\
   -0.5238 &  -0.0319 &   0.7303 &   0.4373\\
    0.0960 &   0.6954 &   0.4363 &  -0.5629 \\
   -0.4277 &   0.6635 &  -0.4951 &   0.3629
\end{array}
\right],
$$
$$
\Sigma =
\left[
\begin{array}{cc}
    4.2674   &      0 \\
         0   &  1.9465 \\
         0   &      0 \\
         0   &      0 
\end{array}
\right],
$$
$$
V =
\left[
\begin{array}{cc}
   -0.4719  &  0.8817 \\
   -0.8817  & -0.4719
\end{array}
\right].
$$
If you need the null spaces then you should use the "full" SVD. However, most problems do not require the "full" SVD.
