I have a question regarding the definition of direct sum of a vector space in relation to subspaces.

Definition: A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1\cap W_2 = \{0\}$ and $W_1 + W_2 = V$. We denote that $V$ is the direct sum of $W_1$ and $W_2$ by writing $V = W_1\oplus W_2$.

Is this definition saying that any vector in $V$ can be written as a linear combination of the vectors in the set $W_1 + W_2$?



The definition is saying that any vector $v \in V$ can be written as $v = w_1 + w_2$ where $w_1 \in W_1$ and $w_2 \in W_2$ (this is the condition $W_1 + W_2 = V$), and this decomposition is unique (this follows from the condition $W_1\cap W_2 = \{0\}$). You don't need to take linear combinations.

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    $\begingroup$ Right, the key word missing from the OPs statement is "unique". Of course what the OP says is true, it does mean any vector can be written as a combination like that. But it's not equivalent to that without specifying "unique". $\endgroup$ – Gregory Grant Feb 14 '16 at 7:24
  • $\begingroup$ Thank you so much guys and thank you for the swift responses! Really cleared up the confusion for me! Thank you for the edit Michael! $\endgroup$ – King Tut Feb 14 '16 at 7:32

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