0

I am really stuck on exercise 5 in chapter 1.3 of Greub's book Multilinear Algebra.

The exercise asks me to show that if $\varphi \colon E \times F \to G$ and $\psi \colon E \times F \to H$ are bilinear maps such that $N_1(\varphi) \subseteq N_1(\psi)$ and $N_2(\varphi) \subseteq N_2(\psi)$, then there exists a linear map $f \colon G \to H$ such that $\psi = f \varphi$.

The definition of $N_1$ and $N_2$ are: $N_1(\varphi) = \{ x \in E \mid \varphi(x, y) = 0 \; \forall y \in F \}$, and $N_2(\varphi) = \{ y \in F \mid \varphi(x, y) = 0 \; \forall x \in E \}$.

It is clear to me that $N_1(\varphi)$ is the kernel of the "adjoint" linear map $E \to \mathrm{Hom}(F,G)$, and $N_2(\varphi)$ the kernel of the map $F \to \mathrm{Hom}(E, G)$. But I don't see how this is helpful.

Another problem is that assuming the exercise to be true I should at the very least be able to show that if $\varphi(x_1, y_1) = \varphi(x_2, y_2)$, for some $(x_1, y_1), (x_2, y_2) \in E \times F$, then $\psi(x_1, y_1) = \psi(x_2, y_2)$ as well. But even this I have no idea how to show.

Any hints or help would be greatly appreciated.

user920957
  • 630
  • 3
  • 14
  • This does not sound right. Consider two symmetric bilinear forms $\varphi$ and $\psi$ on $\mathbb R^2$ defined by $\varphi(x,y)=x^T\pmatrix{1\ &-1}y$ and $\psi(x,y)=x^T\pmatrix{0\ &1}y$. We have $N_i(\varphi)=0\subseteq N_i(\psi)$ for $i=1,2$. Yet, we cannot possibly have $\psi=f\varphi$ for some linear functional $f$, because when $x=(1,1)^T$, we have $\varphi(x,x)=\varphi(0,0)$ but $\psi(x,x)\ne\psi(0,0)$. – user1551 Oct 22 '22 at 17:28
  • @user1551 I posted the correct page as an answer. Preview at https://www.google.com/books/edition/Multilinear_Algebra/jlvoCAAAQBAJ?hl=en&gbpv=1&printsec=frontcover – Will Jagy Oct 22 '22 at 19:08
  • Thank you. That explains why I was having so much trouble solving the exercise. However, now I'm wondering what the author had in mind when he wrote that exercise. – user920957 Oct 22 '22 at 19:35

1 Answers1

2

Different edition from mine...

ERRATA list. Each listing has (p,n) where p is the page number and : postive n means n lines from the top down; n negative means that many lines up from the bottom of the page.

https://github.com/blargoner/math-algebra-greub-errata/blob/master/errata.pdf

enter image description here

enter image description here

Will Jagy
  • 139,541
  • So it is exercise 4 in the first edition then, but exercise 5 in the second edition. – user920957 Oct 22 '22 at 19:17
  • @user, I found an understandable errata list. It is not being kept up by Greub or the publisher, which would be preferred. So, page 4, seven lines up from the bottom, he is saying exercise 5 part b is false – Will Jagy Oct 22 '22 at 20:04
  • Hey that's my errata list! :) Greub has been dead for 30+ years, and I'm pretty confident Springer doesn't care about the errata. – blargoner Oct 22 '22 at 20:17
  • @blargoner thank you. I wasn't sure about Greub. Note that the OP asked in comment for " what the author had in mind when he wrote that exercise," in case you'd like to answer – Will Jagy Oct 22 '22 at 20:20