(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose T : V --> W is a linear transformation and one-to-one. Show, if ||.|| is a norm on W, then ||x|| =||T(x)|| is a norm on V.

(V and W are vector spaces)

2. Relevant equations

T is linear, so T(x+y)= T(x) + T(y) and T(ax)= aT(x)

T is one-to-one, so T(x)=T(y) implies that x=y.

||.|| is a norm, so ||v||=0 iff v=0 and is always greater than or equal to 0;

||cv||=c||v||

||v+w|| is less than or equal to ||v||+||w||

3. The attempt at a solution

I know since T is one-one, then ker(T)={0} and since T is linear, then T(0)=0. I tried using the properties of linear transformations to prove the three properties (listed above) of a norm, and I think it can be solved in this way, but I haven't been able to figure it out.

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# Homework Help: Linear Transformation and Proving Norms

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