Q1
Matrices
Determinant and Rank Computation
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Answer the following questions.
(1) The function $f ( x , y )$ with real variables $x , y$ is defined as follows:
$$f ( x , y ) = \left| \begin{array} { c c c }
1 & x _ { 1 } & y _ { 1 } \\
1 & x _ { 2 } & y _ { 2 } \\
1 & x & y
\end{array} \right|$$
Show that the set of solutions of the equation $f ( x , y ) = 0$ is a line passing through two points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$ on the $x y$ plane, where $x _ { 1 } \neq x _ { 2 }$.
(2) Find the value of the determinant $\left| \begin{array} { c c c } 1 & x _ { 1 } & x _ { 1 } ^ { 2 } \\ 1 & x _ { 2 } & x _ { 2 } ^ { 2 } \\ 1 & x _ { 3 } & x _ { 3 } ^ { 2 } \end{array} \right|$ in factored form.
(3) Show that there is a unique curve $y = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 }$ passing through three points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \left( x _ { 3 } , y _ { 3 } \right)$ on the $x y$ plane, where $a _ { 0 } , a _ { 1 } , a _ { 2 }$ are constants and $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are all distinct.
(4) The curve in (3) can be represented in the form $y = c _ { 1 } y _ { 1 } + c _ { 2 } y _ { 2 } + c _ { 3 } y _ { 3 }$, where each of $c _ { 1 } , c _ { 2 } , c _ { 3 }$ does not depend on $y _ { 1 } , y _ { 2 } , y _ { 3 }$. Find $c _ { 1 } , c _ { 2 } , c _ { 3 }$.
(5) Let us represent a curve $y = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } + a _ { 4 } x ^ { 4 }$ passing through five points $\left( x _ { 1 } , y _ { 1 } \right) , \ldots , \left( x _ { 5 } , y _ { 5 } \right)$ on the $x y$ plane in the form $y = c _ { 1 } y _ { 1 } + \cdots + c _ { 5 } y _ { 5 }$, where each of $c _ { 1 } , \ldots , c _ { 5 }$ does not depend on $y _ { 1 } , \ldots , y _ { 5 }$, and $x _ { 1 } , \ldots , x _ { 5 }$ are all distinct. Find $c _ { 1 }$.