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Let $E$ be a vector space. Let $A,B,C$ be vector subspaces of $E$. Let $\oplus$ denotes the direct sum of vector spaces. Let $\cong$ denotes the vector space isomorphism. If $B \cong C$ then $A \oplus B \cong A \oplus C$. This property has been used in this thread. I would like to ask if the reverse holds, i.e.,

If $A \oplus B \cong A \oplus C$ then $B \cong C$.

Thank you so much for your elaboration!

Analyst
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  • See also this duplicate. It is false. This question has also been asked for abelian groups and modules. – Dietrich Burde May 22 '23 at 11:43
  • @DietrichBurde The link you attached addresses the case $B=C$. Could you send me a link that concerns the case $B \cong C$? – Analyst May 22 '23 at 11:44
  • The first duplicate also answers this for isomorphisms as well. Read the last part of the answer by user14972:" e.g. if $V$ is infinite dimensional, then $$ V \cong V \oplus V \qquad \text{and} \qquad V \cong V \oplus 0 $$ but $V\not\cong 0$. – Dietrich Burde May 22 '23 at 11:50
  • @DietrichBurde Maybe you are using another "direct sum". Mine is the same as the one in the linked Wikipedia page, i.e., "the direct sum of two abelian groups $A$ and $B$ is another abelian group $A \oplus B$ consisting of the ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. To add ordered pairs, we define the sum $(a, b)+(c, d)$ to be $(a+c, b+d)$; in other words addition is defined coordinate-wise." – Analyst May 22 '23 at 11:54
  • @DietrichBurde It seems from here and here that if $\dim E < \infty$ then $\dim B = \dim C$ and thus $B \cong C$. – Analyst May 22 '23 at 11:56
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    Yes, this is the same as for abelian groups. So for finite-dimensional vector spaces the direct sum is "cancellable", but in general not (and this is what you have asked). See this post. – Dietrich Burde May 22 '23 at 12:00
  • @DietrichBurde You are right! Thank you so much! – Analyst May 22 '23 at 12:05

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