Exercise 2.C.13

Question

Suppose $U$ and $W$ are both 4-dimensional subspaces of $C^6$. Prove that there exist two vectors in $U \cap W$ such that neither of these vectors is a scalar multiple of the other.

Arden Shi · Accepted Answer

By 2.43, 
\begin{equation}
\begin{split}
dim (U + W) &= dim U + dim W - dim (U \cap W) \
dim (U + W) &= 8 - dim (U \cap W)
\end{split}
\end{equation}
Now, observe that 
$$
dim (U + W) \leq 6
$$
Therefore, applying it back into the equation,
\begin{equation}
\begin{split}
8 - dim (U \cap W) &\leq 6 \
2 &\leq dim (U \cap W)
\end{split}
\end{equation}

Thus, $\exists$ a basis of length $\geq 2$ with $\geq 2$ vectors that aren't scalar multiples of each other.

Exercise 2.C.13

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Exercise 2.C.13

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