Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I can't find a source online that clearly states the properties of a multilinear function in relation to linear algebra (I say this because I am in an introductory linear algebra class, and this is not included in the textbook). I realized today while studying for the midterm exam tomorrow that I don't know the correct properties of a multilinear function.

Faced with expanding the multilinear function $f(a\vec{e}_1+b\vec{e}_2, c\vec{e}_1+d\vec{e}_2, g\vec{e}_1+h\vec{e}_2 )$ I would have written

$$f(a\vec{e}_1+b\vec{e}_2, c\vec{e}_1+d\vec{e}_2, g\vec{e}_1+h\vec{e}_2 ) = f(a\vec{e}_1, c\vec{e}_1, g\vec{e}_1 ) + f(b\vec{e}_2, d\vec{e}_2, h\vec{e}_2 )=acgf(\vec{e}_1, \vec{e}_1, \vec{e}_1) + bdhf(\vec{e}_2, \vec{e}_2, \vec{e}_2)$$

which is incorrect. I discovered this when looking over the solutions given to an assignment. It seems the correct expansion is

$$f(a\vec{e}_1+b\vec{e}_2, c\vec{e}_1+d\vec{e}_2, g\vec{e}_1+h\vec{e}_2 )=$$

$$acgf(\vec{e}_1, \vec{e}_1, \vec{e}_1) + achf(\vec{e}_1, \vec{e}_1, \vec{e}_2) $$

$$+ adgf(\vec{e}_1, \vec{e}_2, \vec{e}_1) + adhf(\vec{e}_1, \vec{e}_2, \vec{e}_2) $$$$ + bcgf(\vec{e}_2, \vec{e}_1, \vec{e}_1) + bchf(\vec{e}_2, \vec{e}_1, \vec{e}_2) $$$$ + bdgf(\vec{e}_2, \vec{e}_2, \vec{e}_1) + bdhf(\vec{e}_2, \vec{e}_2, \vec{e}_2)$$

What is the procedure to correctly expand a multilinear function as done above? Any help would be appreciated.

share|improve this question
1  
Multlinear is linear with respect to each variable. –  Sigur Feb 5 '13 at 0:14
    
Here's how to think about it "symbolically": $f((ae_1+be_2) \otimes (ce_1+de_2) \otimes (ge_1+he_2))=f(acg e_1 \otimes e_1 \otimes e_1 + \cdots)=acg f(e_1,e_1,e_1)+\cdots$ In fact, this has a name. –  wj32 Feb 5 '13 at 0:19

1 Answer 1

up vote 2 down vote accepted

Let's build up from a multilinear function of two vectors before going to three.

$$f(u + v, w + x) = f(u + v, w) + f(u+v, x)$$

That just exploits linearity in the second argument. Now exploit linearity in the first.

$$f(u+v, w) + f(u+v, x) = f(u, w) + f(v, w) + f(u, x) + f(v, x)$$

Just apply linearity on each separate argument and you should be fine.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.