Principles of Quantum Mechanics · Problem 1.4.2

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · subspaces

Problem

Suppose

\mathbb{V}_1^{n_1}

and

\mathbb{V}_2^{n_2}

are two subspaces such that any element of

\mathbb{V}_1

is orthogonal to any element of

\mathbb{V}_2

. Show that the dimensionality of

\mathbb{V}_1 \oplus \mathbb{V}_2

n_1 + n_2

. (Hint: Theorem 4.)