Exercise 2C Problem 12

Linear Algebra Done Right - Exercise 2C Problem 12 - Solution

June 27, 2022 · 1 min · 77 words · Me | Suggest Changes

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Problem

Suppose $U$ and $W$ are both five-dimensional subspaces of $\mathbb{R}^9$. Prove that $U \cap W \neq \lbrace 0 \rbrace$

Solution

By theorem 2.43, 2.38 and the fact that $U + W$ is a subspace of $\mathbb{R}^9$ by theorem 1.39: $$9 = \dim \mathbb{R}^9 \geq \dim (U + W) = \dim U + \dim W - \dim U \cap W$$ $$= 5 + 5 - \dim U \cap W$$ Implying that $\dim U \cap W \geq 1$.

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