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I'm stuck with the following problem and I've tried approaching it by extending the initial base of $W$ without luck. Any hints??
Consider two subspaces $W_1$ and $W_2$ of the vector space $\mathbb R^2$ such that dim $W_1=\dim W_2=1$. Prove that there exists a subspace $W$ such that $V=W \bigoplus W_1$ and $V=W \bigoplus W_2$.

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What is $V$? $\Bbb R^2$? – David Mitra Jan 26 '12 at 23:46
the problem doesn't specify but I assumed so – bubbly Jan 26 '12 at 23:50
Think about what these spaces really are (as familiar geometric objects). $W_1$ is a line through the origin. $W_2$ is also a line through the origin, maybe the same line as $W_1$, maybe not. Given any two directions in the plane, so long as they aren't parallel or opposite, you can get anywhere in the plane in exactly one way by going an appropriate distance in the first direction, followed by an appropriate distance in the second. – Gerry Myerson Jan 27 '12 at 0:04
Yes I think I should have thought of it that way first instead of rushing to have an algebraic solution. – bubbly Jan 27 '12 at 0:13

Assuming that $V$ is $\Bbb R^2$:

Note that $W_1$ is spanned by one vector, say, ${\bf w_1}\ne\bf 0$ and $W_2$ is spanned by one vector, say, ${\bf w_2}\ne\bf0$.

If $W_1=W_2$, then take any vector $\bf v$ not in $W_1$ and set $W=\text{span}\{{\bf v\}}$ (in this case $\{\bf v, \bf w_1\}$ is a basis of $\Bbb R^2$).

If $W_1\ne W_2$, let $W$ be the span of any vector that is in neither $W_1$ nor $W_2$, say $W =\text{span}\{{\bf w_1+w_2}\}$ (in this case $\{\bf w_1+\bf w_2, \bf w_2\}$ and $\{\bf w_1+\bf w_2, \bf w_1\}$ are bases of $\Bbb R^2$).

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