Exercise-10

LADR

Exercise 1

Suppose \(V_1\) and \(V_2\) are subspaces of \(V\). Prove that the intersection \(V_1 \cap V_2\) is a subspace of \(V\).

Solution

Step-1

\(0 \in V_1\) and \(0 \in V_2\), hence \(0 \in V_1 \cap V_2\).

Step-2

If \(u, v \in V_1 \cap V_2\), then, \(u + v \in V_1 \cap V_2\). This is because \(u, v \in V_1 \implies u + v \in V_1\) and \(u, v \in V_2 \implies u + v \in V_2\).

Step-3

If \(u \in V_1 \cap V_2\), then \(\lambda u \in V_1 \cap V_2\). The argument is similar to the one followed in step-2.

Step-4

Since \(V_1 \cap V_2\) has the identity element of \(V\), is closed under vector addition and scalar multiplication, it is a subspace of \(V\).

Footnotes

  1. Exercise-10, Exercises 1C, Page-24, Linear Algebra Done Right, Fourth Edition, Sheldon Axler↩︎