Homework 4, Math 152-205, additional problem:

The linear system becomes

[ 1  x1  (x1)^2 ] [ a ]     [ y1  ]
[ 1  x2  (x2)^2 ] [   ]     [ y2  ] 
[ ...           ] [ b ]  =  [ ... ]
[ 1  xn  (xn)^2 ] [   ]     [ yn  ]
                  [ c ]

Multiplying by the transpose of the coefficient matrix yields 


[ 1      ... 1      ] [ 1  x1  (x1)^2 ] [ a ]   
[                   ] [ 1  x2  (x2)^2 ] [   ]    
[ x1     ...  xn    ] [ ...           ] [ b ]  =  
[                   ] [ 1  xn  (xn)^2 ] [   ]    
[ (x1)^2 ... (xn)^2 ]                   [ c ]


[ 1      ... 1      ] [ y1 ]
[                   ] [ y2 ]
[ x1     ...  xn    ] [ ...]
[                   ] [ yn ]
[ (x1)^2 ... (xn)^2 ]      

or, in other words

[ n                  x1+...+xn          (x1)^2+...+(xn)^2 ] [ a ]
[ x1+...+xn          (x1)^2+...+(xn)^2  (x1)^3+...+(xn)^3 ] [ b ]  =
[ (x1)^2+...+(xn)^2  (x1)^3+...+(xn)^3  (x1)^4+...+(xn)^4 ] [ c ]

[ y1        + ... + yn        ]
[ x1 y1     + ... + xn yn     ] 
[ (x1)^2 y1 + ... + (xn)^2 yn ]