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let $V=\mathbb{R}^4$ and let $W=\langle\begin{bmatrix}1&1&0&0\end{bmatrix}^t,\begin{bmatrix}1&0&1&0\end{bmatrix}^t\rangle$. we need to find the subspaces $U$ & $T$ such that $ V=W\bigoplus U$ & $V=W \bigoplus T$ but $U\ne T$.

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For example, can you find one space $U$ such that $V=W\oplus U$? – Gerry Myerson Oct 13 '12 at 11:34

HINT: Look at a simpler problem first. Let $X=\{\langle x,0\rangle:x\in\Bbb R\}$, a subspace of $\Bbb R^2$. Can you find subspaces $V$ and $W$ of $\Bbb R^2$ such that $\Bbb R^2=X\oplus V=X\oplus W$, but $V\ne W$?

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