[Math] let A, B and C be subspaces of V . Which of the following statements are true

linear algebravector-spaces

Let $V$ be a finite-dimensional vector space and let $A$, $B$ and $C$ be subspaces of $V$. Which of the following statements are true?

(a) $$A \cap (B + C) = (A \cap B) + (A \cap C)$$
(b) $$A \cap (B + C) \subset (A \cap B) + (A \cap C)$$
(c) $$A \cap (B + C) \supset (A \cap B) + (A \cap C)$$

My attempt:
I was drawing the Venn diagram. From Venn diagram I concluded that
$$A \cap (B + C) = (A \cap B) + (A \cap C)$$ is true ..

Is my answer is correct or not, im not sure about my answer help Me..

Best Answer

In fact, it is only choice c which is correct.

As a counter-example for the other two, consider the following:

  • $V = \Bbb R^2$
  • $A$ is the span of the vector $(1,1)$
  • $B$ is the $x$-axis
  • $C$ is the $y$-axis
Related Question