Problem with notation

[Macmilan]

Hello There :)

I have the following problem: I and my friend are trying to solve the task in the attachment. We are a little bit lost with the notation of g(x) function.

Could anyone help to write function f(x,y) = 3x²y+4xy in notation of g(x)? I think solving this issue might bring some understanding for us so please help.

 

Thank,

Patryk 

[HallsofIvy]

You can't!  The form you give can only be used for quadratic polynomials in x and y while the polynomial $$3x^2y+ 4xy$$ is cubic. If this were simply $3x^2+ 4xy$ then you could. 

Treating (x, y) as the vector $\begin{pmatrix}x \\ y \end{pmatrix}$ and its transpose is $\begin{pmatrix} x & y \end{pmatrix}$ and writing Q as the generic 2 by 2 matrix $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ and the vector a as $\begin{pmatrix}p \\ q\end{pmatrix}$, "$x^TQx+ a^Tx$" becomes $\begin{pmatrix}x & y \end{pmatrix}\begin{pmatrix}a & b \\ c & d \end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}+ \begin{pmatrix}p & q \end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}$$= ax^2+ (b+ c)xy+ dy^2+ px+ qy$.  In order that this equal $3x^2+ 4y$ we must have a= 3, b+ c= 3, d= p= q= 0.  Of course, b+ c= 3 does not tell us either b or c separately.  The simplest thing to do (and what is generally done) is to require that Q be a symmetric matrix so that b= c= 3/2.  Then $Q= \begin{pmatrix}3 & \frac{3}{2} \\ \frac{3}{2} & 0 \end{pmatrix}$.  

Then $3x^2+ 4xy= \begin{pmatrix}x & y \end{pmatrix}\begin{pmatrix}3 & \frac{3}{2} \\ \frac{3}{2} & 0 \end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}$.

Edited April 8, 2019 by HallsofIvy