Is the Direct Sum Complement Unique?

[Bipolarity]

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

 

[mfb]

I am not sure how your direct sum is evaluated. What about this?
V=R²
W={(x,0)}, W₁={(x,x)}, W₂={(x,-x)}

 

[economicsnerd]

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.

 

[Bipolarity]

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

BiP

 

[economicsnerd]

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.