Are there $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus D$, and $A\cong C$, but $B\not\cong D$?

Question

I remember an exercise from Roman's Linear Algebra, but now I can't locate it in the book. Anyway, I think it asked to give examples of $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus D$, and $A\cong C$, but $B\not\cong D$.

I feel like I must be forgetting some additional part of the problem, because the above is impossible, right?

Hint: Consider what happens if the dimensions of $A$, $B$ and $C$ are all countable, but $D$ is finite dimensional. — Tobias Kildetoft, Aug 05 '13 at 19:14
A-ha! Thank you, that has been bothering me in the back of my head for months. — Eric Auld, Aug 05 '13 at 19:15
Does this answer your question? How to find a vector space V and decompositions $V=A\oplus B = C\oplus D$ with $A$ isomorphic to $C$ but $B$ is not isomorphic to $D?$ — BCLC, Mar 06 '20 at 08:48

score 2 · Accepted Answer · answered Aug 05 '13 at 19:18

2

I thought I would post an answer based on Tobias's hint in this community wiki:

If $A, B, C$ are countable-dimensional, but $D$ is finite dimensional, then by cardinal arithmetic, $A \cong A \oplus B \cong C \cong C \oplus D$, but $B \not\cong D$.

answered Aug 05 '13 at 19:18

Eric Auld

28,127

Are there $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus D$, and $A\cong C$, but $B\not\cong D$?

1 Answers1