Find the corresponding vector of a linear functional via $B$-identification.(Bilinear form)

Question

Consider a symmetric matrix $J=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. Consider a induced non-degenerate symmetric bilinear form $B: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$, explicitly I know this metric has the form of $B^J(X,Y) = X^T J Y$.

If we $B$-identify $(\mathbb{R}^2)^*$ with $\mathbb{R}^2$, I want to find the vectors on $\mathbb{R}^2$ which corresponds to a linear functional on $\mathbb{R}^2$ given by $f(x,y) = x+y$.

Answer 1

Hint: The matrix of this functional is $\pmatrix{1 & 1}$. So, you are looking for the vector(s) $X \in \Bbb R^2$ for which $X^TJ = \pmatrix{1 & 1}$.