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 $f$?

To answer your first question:

as it stands: no, because linearity in first variable (with rest variables fixed) doesn't imply linearity in second variable. 

But, $f$ being multilinear means it is linear in each variable (when the rest are fixed), so if we also assume that, the $c\cdot d\cdot$ line is correct.

For the actual question, I couldn't find simpler solution for now, than to introduce 8 undeterminates, say $a_1,..,a_8$, the coefficients of the appropriate "matrices" where $f$ is given.
So that $a_1$ belongs to the matrix $((1,0),(1,0),(1,0))$, that is, we need
$$a_1((1,0),(1,0),(1,0)) +a_2((1,0),(1,0),(0,1))+... = ((2,3),(−1,1),(7,4)) $$
that is,
$$\begin{align} a_1+a_2+a_3+a_4 &= 2  \\
a_5+a_6+a_7+a_8 &=3 \\
a_1+a_2+a_5+a_6 & = -1 \\
 \dots
\end{align} 
$$
Then you can solve it using Gaussian elimination, for example. Hmm.. having 6 equations for 8 variables. But it usually means that you can choose 2 variables freely in the solution...
