Page 624 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Consequently,

Solution (b)

If

(6)

is any vector in , then from the given formula for T,

To find and (7)
, we must express 6 and 7 as linear combinations of and . This yields the vector equations

(8)

Equating corresponding entries yields the linear systems (9)
and (10)
Solving 10 for and yields (11)
so

and solving 11 for and yields
so

Thus

so 5a holds.

Matrices of Identity Operators

The matrix for the identity operator on V always takes a special form.