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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q12E (page 199)

State true or false, if Tis an isomorphism then $T^{- 1}$ +must be an isomorphismas well.

Short Answer

Expert verified

The statement is true.

Step by step solution

01

Determine the linearity of T-1.

Consider the function T such that T is an isomorphism.

AsT is an isomorphism means $\dim \{k e r (T)\}$ is $\{0\}$ and inverse of the function is exist and linear.

By the property of linear transformation, the function $T^{- 1}$ is linear.

02

Determine whether the statement is True or False

As inverse of the function is exist means for every x there exist y such that .

${(T^{- 1})}^{- 1} (x) = y$

Simplify the equation ${(T^{- 1})}^{- 1} (x) = y$ as follows.

$\begin{array}{rcl} {(T^{- 1})}^{- 1} (x) & = & y \\ T (x) & = & y \end{array}$

As the function $T^{- 1}$ is exist and linear, by the definition of isomorphism, the function $T^{- 1}$ is an isomorphism.

Hence, the statement is true.

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