# Homework Help: Linear Transformation and Magnitudes

1. Feb 28, 2009

### Nebula

1. The problem statement, all variables and given/known data
From Calculus on Manifolds by Spivak: 1-10
If T:Rm -> Rn is a Linear Transformation show that there is a number M such that |T(h)| $$\leq$$ M|h| for h$$\in$$Rm

2. Relevant equations
T is a Linear Transformation
=> For All x,y $$\in$$ Rn and scalar c
1. T(x+y)=T(x)+T(y)
2. T(cx)=cT(x)

3. The attempt at a solution
Well I didn't get very far but I do know this. The matrix of T with respect to the standard basis is A. Ah=T(h)
So we can write what |T(h)| and |h| look like.
|T(h)|=sqrt((a11h1+...+a1nhn)2+...+(am1h1+...+amnhn)2)
where aij's are the elements in the matrix representation of T, A.

|h|=sqrt((h1)2+...+(hn)2)

Suggestions, hints and clues are all appreciated!

2. Mar 1, 2009

### yyat

Try to show that the function $$f:\mathbb{R}^m\to\mathbb{R}$$, $$f(x)=|T(x)|$$ has a maximum when restricted to the unit sphere $$S^{m-1}$$ in $$\mathbb{R}^m$$. Then use the fact that every nonzero vector can be written in the form $$\lambda v$$, where $$\lambda$$ is a scalar and $$v$$ is a unit vector.