# Classifying PDE and transforming to real inner space from bilinear def

$$Φ(x, y) = 5x1x2−x1y2−x2y1+5y1y$$

$$∀ x = ( x1, y1 )^T ∈ R^ 2$$

$$∀ y = ( x2, y2 )^T ∈ R^ 2$$

We need to show that it's (1) symmetric (2)quadratic form (3) classify it (4) there is this one question asking if it is possible to use the bilinear form $$Φ$$ to turn the vector space $$(R^ 2 , R)$$ into an real inner product space? If yes, explain why and how this can be achieved

From My understanding in order to find

1) symmetric

we will have to come up with this $$Φ(x,y)=Φ(y,x)$$ which can be referred to as a matrix representation ( But I'm quite not sure what will be that matrix be )

My guess is it will be the matrix

[ -1 5, 5 -1 ] But not sure

2) the quadratic form will be indefinite

and would really appreciate helping me with the rest

• Use an underscore for a subscript i.e u_{i} encased in dollar signs gives $u_{i}$. – mattos Feb 18 at 2:57