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CSS Pure Mathematics QUESTION #4132
Question 1
For subspaces $W_1, W_2$ of $\mathbb{R}^3$ with $\dim(W_1) = 2$ and $\dim(W_2) = 2$, what is the minimum $\dim(W_1 \cap W_2)$?
  • $0$
  • $1$✔️
  • $2$
  • $4$
Correct Answer Explanation
By dimension formula: $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$. Since $\dim(W_1 + W_2) \leq 3$, we have $2 + 2 - \dim(W_1 \cap W_2) \leq 3$, so $\dim(W_1 \cap W_2) \geq 1$.