Abstract Algebra: Theory and Applications · Problem 20.17

Fields — Thomas Judson

Mathematics — Abstract Algebra · direct-sums

Problem

Direct Sums. Let

U

and

V

be subspaces of a vector space

W

. The sum of

U

and

V

, denoted

U + V

, is defined to be the set of all vectors of the form

u + v

, where

u \in U

and

v \in V

. (a) Prove that

U + V

and

U \cap V

are subspaces of

W

. (b) If

U + V = W

and

U \cap V = \mathbf{0}

, then

W

is said to be the direct sum. In this case, we write

W = U \oplus V

. Show that every element

w \in W

can be written uniquely as

w = u + v

, where

u \in U

and

v \in V

. (c) Let

U

be a subspace of dimension

k

of a vector space

W

of dimension

n

. Prove that there exists a subspace

V

of dimension

n - k

such that

W = U \oplus V

. Is the subspace

V

unique? (d) If

U

and

V

are arbitrary subspaces of a vector space

W

, show that

\dim(U + V) = \dim U + \dim V - \dim(U \cap V).

Hint. (a) Let

u, u' \in U

and

v, v' \in V

. Then

(u + v) + (u' + v') = (u + u') + (v + v') \in U + V

\alpha(u + v) = \alpha u + \alpha v \in U + V.