# Direct sum complement is unique

## Main Question or Discussion Point

I'm curious about whether a statement I conjecture about direct sums is true.
Suppose that $V$ is a finite-dimensional vector space and $W$,$W_{1}$,$W_{2}$ are subspaces of $V$. Let $V = W_{1} \bigoplus W$ and $V = W_{2} \bigoplus W$.

Then is it the case that $W_{1} = W_{2}$?

I merely need to know whether this is true or not so that I can know which direction to steer my proof. I am guessing it is true, but am having trouble proving it, and that is giving me doubts as to whether or not it is true.

All help is appreciated! Thanks!

BiP

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mfb
Mentor
V=R2
W={(x,0)}, W1={(x,x)}, W2={(x,-x)}

In the category of vector spaces: no, as mfb showed.
In the category of inner-product spaces: yes. In this case, we say $V=W_i\oplus W$ if $V=W_i+W$ and $W_i, W$ are orthogonal.

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Thanks a lot guys! No wonder my proof has not been working out!
How might I prove that orthogonal complements are unique?

BiP

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Suppose $V=W_1+ W$ and $W_1 \perp W$. Let $W_2 = \{v\in V: \enspace v\perp W\}$.

By construction, $W_2 \supseteq W_1$. Try to show that $W_1, W_2$ have the same (finite) dimension... hint: dimension theorem. Then use that no finite-dimensional vector space has a proper subspace of the same dimension.