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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
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:
View attachment 9392
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
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:
View attachment 9392
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
