Write as a Vector Equality x^2-xy+y^2=21 , x^2-2xy-8y^2=0

Problem

{[x2−x*y+y2=21],[x2−2*x*y−8*y2=0])

Solution

1. Identify the variables and the structure of the system of equations to represent it using vector and matrix notation.

2. Define the vector of variables x as a column vector containing x and y

x=[[x],[y]]

3. Express the quadratic forms using symmetric matrices (A_1) and (A_2) such that each equation follows the form xT*(A_i)*x=(c_i)

4. Determine the matrix (A_1) for the first equation x2−x*y+y2=21 by splitting the x*y coefficient equally between the off-diagonal elements.

(A_1)=[[1,−0.5],[−0.5,1]]

5. Determine the matrix (A_2) for the second equation x2−2*x*y−8*y2=0 using the same method.

(A_2)=[[1,−1],[−1,−8]]

6. Combine the equations into a single vector equality where the function ƒ(x) maps the vector x to the vector of constants.

ƒ(x)=[[xT*(A_1)*x],[xT*(A_2)*x]]=[[21],[0]]

Final Answer

[[[[x,y]]*[[1,−0.5],[−0.5,1]]*[[x],[y]]],[[[x,y]]*[[1,−1],[−1,−8]]*[[x],[y]]]]=[[21],[0]]

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