Linear Algebra Done Right · Problem 3E.1

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · products-vector-spaces

Problem

Suppose

T

is a function from

V

W

. The graph of

T

is the subset of

V \times W

defined by

\text{graph of } T = \{(v, Tv) \in V \times W : v \in V\}.

Prove that

T

is a linear map if and only if the graph of

T

is a subspace of

V \times W

. Formally, a function

T

from

V

W

is a subset

T

V \times W

such that for each

v \in V

, there exists exactly one element

(v, w) \in T

. In other words, formally a function is what is called above its graph. We do not usually think of functions in this formal manner. However, if we do become formal, then this exercise could be rephrased as follows: Prove that a function

T

from

V

W

is a linear map if and only if

T

is a subspace of

V \times W

Related theory

Tangent Spaces And Vector Fields

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