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 $V=W \bigoplus W_2$.
Vector subspace decomposition problem (Linear algebra)
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linear-algebra
vector-spaces
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0Yes I think I should have thought of it that way first instead of rushing to have an algebraic solution. – 2012-01-27
1 Answers
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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 $\Bbb R^2$).