16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 2.2.5

Vector Spaces

Mathematics — Linear Algebra · subspaces-of-matrix-spaces

Problem

Let

F

be a field and let

n

be a positive integer (

n \geq 2

). Let

V

be the vector space of all

n \times n

matrices over

F

. Which of the following sets of matrices

A

V

are subspaces of

V

? (a) all invertible

A

; (b) all non-invertible

A

; (c) all

A

such that

AB = BA

, where

B

is some fixed matrix in

V

; (d) all

A

such that

A^2 = A