Let $f:\mathbf{R}^2\to\mathbf{R}^2$, $f\left(\begin{bmatrix}x_1 \\ y_1\end{bmatrix},\begin{bmatrix}x_2 \\ y_2\end{bmatrix}\right) = 2x_1x_2 + 3x_1y_2 + x_2y_1$. As an example we want to represent $f$ with the standard basis. It follows that: $$f(e_1,e_1)=2 \ ;\ f(e_1,e_2)=3 \ ; \ f(e_2,e_1) = 1 \ ; \ f(e_2,e_2)=0 $$
Hey,
Can someone explain to me how e.g. $f(e_1, e_1) = 2$, according to the definition of bilinear forms and matrices? I think i am confused about the indices. I do not really understand what $e_1$ is referring to: is it the first row of the identity-matrix?
I added the example of my class and my question as a picture (i dont know how to write mathematical expression on my computer)
Q:
- What does $f(e_1,e_1)$ mean?
- How is it related to the theorem of bilinear forms: $$\boxed{f(v,w) = v\ ^t\!Aw = \sum_{i=1}^n\sum_{j=1}^n a_{ij}v_iw_j}$$
As you will see, $\beta=\{e_1,e_2\}$ is a basis for $R^2$. So, what you define bilinear form $f$ as $[f]_\beta$ and correspondingly you find the respective coordinates of the matrix that is, for example, You say $f(e_1,e_1)$ is first element of first row and you calculated that $f(e_1,e_1)=2$. Likewise, you calculate other entries!! Hope it helps!!