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

Linear Algebra Done Right

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

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