What Is a Multilinear Function in Multilinear Algebra?

[BrainHurts]

I'm reading Lee's Introduction to Smooth Manifolds and I have a question on the definition of a multilinear function

Suppose $V_{1},...,V_{k}$ and $W$ are vector spaces. A map $F:V_{1} \times ... \times V_{k} \rightarrow W$ is said to be multilinear if it is linear as a function of each variable separately when the others are held fixed: for each i,

$F(v_{1},...,av_{i} + a'v^{'}_{i},...,v_{k}) = aF(v_{1},...,v_{i},...,v_{k}) + a'F(v_{1},..., v^{'}_{i},...,v_{k})$

I'm thinking that it should look like this,

$F(u_{1},...au_{k} + a'v_{1},...,v_{k}) = aF(u_{1},...,u_{k})+a'F(v_{1},...,v_{k})$
any comments?

 

[Office_Shredder]

F on the left hand side takes in 2k-1 vectors, and on the right hand side just k.

The definition in the book is correct, do you think it's a typo or are you just throwing out an idea for a different definition?

 

[BrainHurts]

different idea of course sorry, definitely not trying to correct him, I'm not seeing it as clearly, umm "F on the left hand side." Which F are you talking about? Lee's definition or what I'm thinking it should look like?

 

[Office_Shredder]

OK just wanted to make sure. The F on the left hand side I'm talking about here.

$F(u_{1},...au_{k} + a'v_{1},...,v_{k}) = aF(u_{1},...,u_{k})+a'F(v_{1},...,v_{k})$
any comments?

Let's suppose k=2. It looks like what you have written is

$$F(u_1, a u_2 + a' v_1, v_2) = a F(u_1, u_2) + a' F(v_1, v_2)$$

And F on the left hand side of the equation, and on the right hand side of the equation, have a different number of variables as input

 

[BrainHurts]

Hmm, honestly I'm still a little confused, I'm trying to think of a more concrete example such as F being an inner product

if we let V=ℝⁿ for example, then V is a vector space over the field ℝ with the usual addition of vectors and scalar multiplication.

so $<\cdot,\cdot> : V \times V \rightarrow ℝ$ given by $\sum_{i=1}^{n} a_{i}b_{i}$ is a multilinear function (namely a bilinear function)

so if we let x,y,z, be in ℝⁿ and let F = <.,.>, then F(ax+by,z) = aF(x,z) + bF(y,z)
= a<x,z> + b<y,z> right?

so using Lee's notation I'm not really seeing it.

 

[BrainHurts]

or is it better to see it this way?

$F(a(u_{1},...,u_{k}) + b(v_{1},...,v_{k})) = aF(u_{1},...,u_{k}) + bF(v_{1},...,v_{k})$ ?

 

[micromass]

or is it better to see it this way?

$F(a(u_{1},...,u_{k}) + b(v_{1},...,v_{k})) = aF(u_{1},...,u_{k}) + bF(v_{1},...,v_{k})$ ?

No, that would imply that $F$ is linear.

 

[adriank]

In the case k = 2, the definition says that
$$F(a v_1 + a' v_1', v_2) = a F(v_1, v_2) + a' F(v_1', v_2)$$
and
$$F(v_1, a v_2 + a' v_2') = a F(v_1, v_2) + a' F(v_1, v_2').$$

In other words, the map $V_1 \to W,\;v_1 \mapsto F(v_1, v_2)$ is linear (for fixed $v_2$), and the map $V_2 \to W,\;v_2 \mapsto F(v_1, v_2)$ is linear (for fixed $v_1$). This is what it means to be linear as a function of each variable separately.

 

[BrainHurts]

In the case k = 2, the definition says that
$$F(a v_1 + a' v_1', v_2) = a F(v_1, v_2) + a' F(v_1', v_2)$$
and
$$F(v_1, a v_2 + a' v_2') = a F(v_1, v_2) + a' F(v_1, v_2').$$

In other words, the map $V_1 \to W,\;v_1 \mapsto F(v_1, v_2)$ is linear (for fixed $v_2$), and the map $V_2 \to W,\;v_2 \mapsto F(v_1, v_2)$ is linear (for fixed $v_1$). This is what it means to be linear as a function of each variable separately.

got it! thanks!