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\title{ 5710 Fall 2013. Home Work assignment 1}
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1. Conside an equation $A x=b$
 where 
$$A=
\left( \begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 2 & 0 & 0\\
-1 & 1 & 2 & 2\\
3 & 0 & 1 & 1
\end{array} 
\right), \quad b=
\left( \begin{array}{llll}
1  \\
0 \\
2 \\
1 
\end{array} 
\right),
$$
Show that $A^{-1} $ does not exists, find the least square solution with the regilarizing term $ \vareps \| x-a \|^2, $a^T=(1,\. 0)$.

2. Consider a quadratic form $Q=x^T A X$ where 
$$
A=
\left( \begin{array}{llll}
1 & 1 & 0 & 0 \\
1 & 2 & 1 & 0\\
0 & 1 & 2 & 1\\
0 & 0 & 1 & 2
\end{array} 
\right)
$$
Represent it as a sum of squares $$ Q=\sum_{i=1}^4 c_i y_i^2$$ where $y=Lx$ is a linear transformation of the coordinates, and $L$ is a triangular matrix. Is the form positively defined? 


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