Multilinear Functions I have a question regarding the properties of a multilinear function. This is for a linear algebra class. I know that for a multilinear function,
$f(cv_1,v_2,\dots,v_n)=c⋅f(v_1,v_2,\dots,v_n)$
Does this imply
$f(cv_1,dv_2,\dot,v_n)=c⋅d⋅f(v_1,v_2,\dots,v_n)$?
It is for a question involving a multilinear function $f:\mathbb{R}^2\times\mathbb{R}^2\times\mathbb{R}^2\to \mathbb{R}$. I am given eight values of $f$, each of which is composed of a combination three unit vectors. They are:
$f((1,0),(1,0),(1,0))=e$  
$f((1,0),(1,0),(0,1))=\sqrt{7}$
$f((1,0),(0,1),(1,0))=0$
$f((1,0),(0,1),(0,1))=2$
$f((0,1),(1,0),(1,0))=\sqrt{5}$ 
$f((0,1),(1,0),(0,1))=0 $
$f((0,1),(0,1),(1,0))=\pi$  
$f((0,1),(0,1),(0,1))=3$
Then I am asked to compute for different values of f. For instance,
$f((2,3),(-1,1),(7,4))$
How do I approach this question to solve for the value of $f$?

A multilinear function $f: V_1 \times \ldots \times V_n \to W$, where all the $V_i$ and $W$ are vector spaces is a function such that if we fix $v_1, \ldots v_{i-1}, v_{i+1}, \ldots v_n$, the function $\phi(v_i) := f(v_1, \ldots, v_i, \ldots, v_n)$ is linear in $v_i$.  Linearity means:
$$
\forall v,w \in V_i, a \in \mathbb{R}: \phi(a v + w) = a \phi(v) + \phi(w)
$$
Now, since we know that your function is multilinear, and we know it's value for all possible combinations of linearly independent vectors in $V_1, V_2$ and $V_3$, we can express every input $(v_1, v_2, v_3)$ as linear combinations of the unit vectors in the various spaces and then use the multilinearity to calculate the correct value.
For example:
$$
((2,3),(-1,1),(7,4)) = (2(1,0)+3(0,1), -1(1,0)+(1,0), 7(1,0)+4(0,1)).
$$
I hope this clears it up. ;)
