Linear Transformation and Magnitudes

[Nebula]

Homework Statement

From Calculus on Manifolds by Spivak: 1-10
If T:R^m -> Rⁿ is a Linear Transformation show that there is a number M such that |T(h)| \leq M|h| for h\inR^m

Homework 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)

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+...+a_1nhⁿ)²+...+(a_m1h¹+...+a_mnhⁿ)²)
where a_ij's are the elements in the matrix representation of T, A.

|h|=sqrt((h¹)²+...+(hⁿ)²)

Suggestions, hints and clues are all appreciated!

 

[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.