A First Course in Linear Algebra · Problem T30

Vector Spaces — Robert Beezer

Mathematics — Linear Algebra · vector-space-scalar-one

Problem

Suppose that

V

is a vector space and

\alpha \in \mathbb{C}

is a scalar such that

\alpha \mathbf{x} = \mathbf{x}

for every

\mathbf{x} \in V

. Prove that

\alpha = 1

. In other words, Property O is not duplicated for any other scalar but the ``special'' scalar, 1. (This question was suggested by James Gallagher.)