# Matrix Representation of a Bilinear Form

Let $b : \Bbb{R}^2 \times \Bbb{R}^2 \to \Bbb{R}$ be the bilinear form defined by $$b((x_1,x_2),(y_1,y_2)) = x_1 y_1 - 2x_1 y_2 + x_2 y_1 + 3x_2 y_2.$$ Find the $2 \times 2$ matrix $B$ of $b$ relative to the basis $U=\{u_1,u_2\} = \{(0,1),(1,1)\}$

From what I understand, if $B=[b_{ij}]$ is the matrix representation, then $b_{ij} = b(u_i,u_j)$. If this is so, then I should get $$b_{11} = b((0,1),(0,1)) = 0 \cdot 0 - 2 \cdot 0 \cdot 1 + 1 \cdot 0 + 3 \cdot 1 \cdot 1 =3,$$ $$b_{12} = b((0,1),(1,1)) = 4,$$ $$b_{21} = b((1,1),(0,1)) = 1,$$ and $$b_{22} = b((1,1),(1,1)) = 3$$

However, the answer is $B = \begin{bmatrix} 0 & 4 \\ -1 & 3 \\ \end{bmatrix}$. Where did I go wrong?

If a vector has coordinates $[a,b]$ in $U$, then the vector is $a(0,1)+b(1,1)=(b,a+b)$.
Let's compute $b$ in coordinates in $U$. I will be using $[\cdot,\cdot]$ to denote coordinates in $U$ and $(\cdot,\cdot)$ to denote the vectors themselves or, since they look the same, coordinates in $\{(1,0),(0,1)\}$
• @user193319 I computed the matrix of the bilinear form in the basis $U$. That matrix is, by definition, the matrix such that by doing matrix multiplication with the coordinates of the input in that basis, gives you the value of the form. Aug 3, 2017 at 22:38
Are you sure you've written the bilinear form correctly? If instead it is $$b((x_1,\boldsymbol{y_1}),(\boldsymbol{x_2},y_2))=x_1y_1−2x_1y_2+x_2y_1+3x_2y_2,$$ then you will get the answer you're looking for.