- #1

noelo2014

- 45

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## Homework Statement

I've been stuck on this problem for a while, I actually have the answer (found it in my book), but I'm having trouble getting my head around the concept.

The question is:

Given a linear transformation T:V->W prove that kernel(T) : {vεV : T

_{V}=0

_{W}}

is a subspace of V

## Homework Equations

## The Attempt at a Solution

From what I understand the Kernel is defined as the set of vectors in V that map onto the zero vector in W when transformed under T

Also a subspace W of a vector space V is defined as 1 or more vectors from V that, when added together or scalar multiplied, produce another vector "in" W. (I'm not quoting my book exactly)

Ok, the book gives the proof for this as something like:

*To show that ker(T) is a subspace we must show that it contains at least one vector and is closed under addition and scalar multiplication.*

Since ker(T) contains 0

T(u+v)=T(u)+T(v)=0+0=0

and

T(cu)=T(0c)=T(0)=0

c.T(u)=0

∴ T(cu)=T(u)

Since ker(T) contains 0

_{V}thenT(u+v)=T(u)+T(v)=0+0=0

and

T(cu)=T(0c)=T(0)=0

c.T(u)=0

∴ T(cu)=T(u)

which proves ker(T) is a subspace of V (although I don't understand this proof)

What I'm trying to understand is : If ker(T) refers to a set of vectors in V, isn't that enough proof in itself? Isn't it just like saying "Prove that a subset of V is a subspace of V?" In fact how would I prove that? Still confused about subspaces.