# Thread: Direct Sum & Linear Mappings

1. ## Direct Sum & Linear Mappings

Hi there,

I was just wondering if anyone could tell me if my answer was somewhat right.

An example in my textbook asks:

If T:V X V $\displaystyle \rightarrow$ V is the linear mapping T(x,y)=x-y, determine the kernel and range of T.

Let a=(x,y)
Ker T = {a $\displaystyle \in$ V X V | Ta=0}

0 =Ta
=T(x,y)
= x-y
y = x

Therefore, Ker T = {(x,y)$\displaystyle \in$ V X V | x=y}

Im T = T(V X V) = {Ta | a $\displaystyle \in$ V}
= T(x,y)
= x - y $\displaystyle \forall x,y \in V$

Make sense?

2. Originally Posted by manygrams
Hi there,

I was just wondering if anyone could tell me if my answer was somewhat right.

An example in my textbook asks:

If T:V X V $\displaystyle \rightarrow$ V is the linear mapping T(x,y)=x-y, determine the kernel and range of T.

Let a=(x,y)
Ker T = {a $\displaystyle \in$ V X V | Ta=0}

0 =Ta
=T(x,y)
= x-y
y = x

Therefore, Ker T = {(x,y)$\displaystyle \in$ V X V | x=y}

Im T = T(V X V) = {Ta | a $\displaystyle \in$ V}
= T(x,y)
= x - y $\displaystyle \forall x,y \in V$

Make sense?

The first part does make much sense, the second one much less: the image is the whole of V (why?)

Tonio