How is $ f((x_1.y_1),(x_2,y_2)) = x_1y_1 + x_2y_2$ a bilinear form?

Question

In section $10.1$ of Hoffman-Kunze's Linear Algebra, exercise $2$ states the following:

Let f be the bilinear form on $\mathbb{R}^2$ defined by $f((x_1,y_1),(x_2,y_2)) = x_1y_1 + x_2y_2.$ Find the matrix of f in each of the following bases:

$$\{(1,0),(0,1)\}, \{(1, -1), (1, 1)\}, \{(1,2), (3,4)\}$$

Now, I don't think that is a bilinear form, because $c*f((x_1,y_1),(x_2,y_2)) = cx_1y_1 + cx_2y_2$, while $f(c*(x_1,y_1),(x_2,y_2)) = c^2x_1y_1 + cx_2y_2$.

Answer 1

You are correct; that must be a typo in the exercise. Probably what was meant was either $$ f(\langle x_1,y_1\rangle,\langle x_2,y_2\rangle) = x_1x_2 + y_1y_2 $$ or $$ f(\langle x_1,y_1\rangle,\langle x_2,y_2\rangle) = x_1y_2 + y_1x_2 $$