I was given this question in a linear algebra assignment It tells us that V is a vector space over an infinite field F and $ W \subset V $ is a non trivial subspace of V (neither $ V $ nor the the zero space).
I am asked to prove that there exist an infinite number of different "direct sum complements" for $ W $, but I don't understand why the claim is true, for example the vector space $ V=R^3 $ over $ F = R $ and its subspace $ W = \{ (x,y,0) | x,y \in R \} $ has only one direct sum complement which would be $ U = \{ (0,0,z) | z\in R \} $ not an infinite number as stated in the exercise. What am I missing and what is the correct approach? Thank you all
An idea: find a basis for $\;W\;$ , say $\;B_W:=\left\{\,(1,0,0)\,,\,\,(0,1,0)\,\right\}\;$ . Now, complete this to a basis $\;B_V\;$of the whole $\;\Bbb R^3\;$ . Each such completing of the basis will give you a different direct complement.
For example, if you add the vector $\;(1,1,1)\;$ to $\;B_W\;$ , you get a basis $\;B_V\;$ and Span$\,\{(1,1,1)\}\;$ is a direct complement of $\;W\;$ . Likewise , you could have chosen one of the vectors $\;(0,0,1)\,,\,\,(1,-1,2)\,,\,\,(1,0,1)\,$ , etc., and each time you get a different direct complement to $\;W\;$