# Thread: Question on linear transformation in analysis

1. ## Question on linear transformation in analysis

Hi everybody
I am stuck on this question and hope to get some ideas from u guys
(a) If T: R^m --> R^n is a linear transformation, show that there is a number M such that |T(h)|=< M|h| for h in R^m
(b) Prove that T is continuous

Here is my attempt:
T: R^m --> R^n is a linear transformation if:

T(x+y) = T(x)+T(y) for all x, y in R^m
T(kx) = kT(x) for k is real number

What do I have to do now? and how do you start part b?

Thanks for any suggestions

2. Originally Posted by knguyen2005
Hi everybody
I am stuck on this question and hope to get some ideas from u guys
(a) If T: R^m --> R^n is a linear transformation, show that there is a number M such that |T(h)|=< M|h| for h in R^m
(b) Prove that T is continuous

Here is my attempt:
T: R^m --> R^n is a linear transformation if:

T(x+y) = T(x)+T(y) for all x, y in R^m
T(kx) = kT(x) for k is real number

What do I have to do now? and how do you start part b?

Thanks for any suggestions
Let $\displaystyle e_{k=1,\ldots, m}$ be a base of $\displaystyle \mathbb{R}^m$ and let $\displaystyle h=\sum_{k=1}^m h_k e_k$. Then you have (by linearity of T, the triangle inequality, and because $\displaystyle |h_k|\leq |h|$ for all $\displaystyle k=1,\ldots, m$):
$\displaystyle |T(h)| = |\sum_{k=1}^m h_k T(e_k)|\leq \sum_{k=1}^m |h_k| \,|T(e_k)|\leq |h| \, \sum_{k=1}^m|T(e_k)|= M \, |h|$
where $\displaystyle M := \sum_{k=1}^m|T(e_k)|$