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&#039; v_1&#039;, v_2) = a F(v_1, v_2) + a&#039; F(v_1&#039;, v_2)
and
F(v_1, a v_2 + a&#039; v_2&#039;) = a F(v_1, v_2) + a&#039; F(v_1, v_2&#039;).

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&#039; v_1&#039;, v_2) = a F(v_1, v_2) + a&#039; F(v_1&#039;, v_2)
and
F(v_1, a v_2 + a&#039; v_2&#039;) = a F(v_1, v_2) + a&#039; F(v_1, v_2&#039;).

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!