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I'm given this test assignment, and I can't decipher what it says. Would you kindly help me?

Here's the assignment itself:

Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U \oplus W = V$. Let $S \subseteq U$ and $T \subseteq W$ be two finite linearly independent sets. Prove that $S \cup T$ is linearly independent.

I don't think that ring-sum symbol here is actually the ring sum (xor, exclusive or). Because, if it was, that would mean that either $V$ or $W$ didn't include the trivial (zero) subspace, which would disqualify them from being (sub-)spaces.

So, could you guess what could be meant by this symbol?

rschwieb
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wvxvw
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3 Answers3

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The symbol $\oplus$ denotes the direct sum of (here) vector spaces; in this context, as $U,W$ are said to be subspaces of $V$, it denotes the internal direct sum, i.e., $U\cap W=\{0\}$ and $U,W$ together span $V$.

Hagen von Eitzen
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It means the direct sum. That is

$$U \oplus W = \{u+ w ; u \in U \ \ \text{and}\ \ w\in W \}$$

and $$U \cap W = \{0\}$$

Aaron Maroja
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The symbol $\oplus$ means direct sum. You can look at any linear algebra book for definition. Here is one link: http://www.math.ucla.edu/~pskoufra/M115A-DirectSumOfVectorSpaces.pdf You can also see Direct Sum of vector subspaces

Community
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Krish
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