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- #1

- Jun 22, 2012

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I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding the concepts in Proposition 8.6 ...

Proposition 8.6 reads as follows:

In the above proposition, Browder defines the distance function \rho (S, T) as follows:

\(\displaystyle \rho (S, T) = \| S - T \| \)

... but just some basic questions ...

How do we define \(\displaystyle -T\)?

Is \(\displaystyle -T = ( -1) T\)?

Is \(\displaystyle \| -T \| = \| T \| \)?

A simple example that shows the way things work as I see it follows:

Consider \(\displaystyle T: \mathbb{R}^2 \to \mathbb{R}^2 \)

Let \(\displaystyle T(x,y) = ( x - y, 2y )\)

... then ...

\(\displaystyle - T (x,y) = (-1) T(x,y) = ( -x + y, -2y)\)

and then it follows that ...

\(\displaystyle \| T(x,y) \| = \| ( x - y, 2y ) \| = \sqrt{ (x - y)^2 + (2y)^2 }\)

and ...

\(\displaystyle \| -T(x,y) \| = \| ( -x + y, -2y) \| = \sqrt{ (-x + y)^2 + (-2y)^2 } = \| T(x,y) \| \)

Is the above example correct?

Peter