Linear Algebra Done Right 2nd Edition – Chapter 1 Exercise 8

Chapter 1 Exercise 8.
Prove the intersection of any collection of subspaces of V is a subspace of V .

Proof:
Let $X_1, X_2,...$ be subspaces of $V$ .
Then $O_V \in X_q \forall q$ . Thus,

(1) $O_V \in X_1 \cap X_2 \cap ...$

Therefore, $X_1 \cap X_2 \cap ... \neq \emptyset$ .
Suppose $X_1 \cap X_2 \cap ... = \{ x_1, x_2, ... \}$ . Then $x_i, x_j \in X_q \forall i, j, q$ .
Since $X_q$ is a subspace, $x_i + x_j \in X_q \forall q$ . Thus

(2) $x_i + x_j \in X_1 \cap X_2 \cap ...$

Since $X_q$ is a subspace, $ax_i \in X_q \forall q, i$ and scalar $a$ . Thus

(3) $ax_i \in X_1 \cap X_2 \cap ...$

By 1, 2 and 3, $X_1 \cap X_2 \cap ...$ is a subspace of $V$ .

$\square$

Dedra L. Welch

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Linear Algebra Done Right 2nd Edition – Chapter 1 Exercise 8