Kunze Linear Algebra · Problem 2.3.10

Vector Spaces

Mathematics — Methods · finite-dimensional-spaces

Problem

Let

V

be the set of all 2 × 2 matrices

A

with complex entries which satisfy

A_{11} + A_{22} = 0

. (a) Show that

V

is a vector space over the field of real numbers, with the usual operations of matrix addition and multiplication of a matrix by a scalar. (b) Find a basis for this vector space. (c) Let

W

be the set of all matrices

A

V

such that

A_{21} = -\overline{A_{12}}

(the bar denotes complex conjugation). Prove that

W

is a subspace of

V

and find a basis for

W

Related theory

Dimensional Formulae And Dimensional Equations

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