Question

Let V be any vector space, and let T: V to V be defined by T(v)=3v.
(a) What is the kernel of T?
(b) What is the range of T?

Answer 1

b

Answer 2

lol

Answer 3

T(0)=3*0
v=0 is the kernel right?
is that how it goes?

Answer 4

thats right :)

Answer 5

:)

Answer 6

so how do we do the range?
i don't think this range is the same as the algebraic range right?
or did i just sound retarded?

Answer 7

The kernel of T is the set of vectors v such that T(v) = 0. Since, in this case, T(v) = 3v, this will only be satisfied by v = 0. The range of T is the set of all vectors v such that T(u) = v for some input u. So then the range will be \[v | (1/3)*v \in V\]

Answer 8

ok so you just find the inverse

Answer 9

And it is in a way the same thing as the algebraic range - in fact, the algebraic range is just the 2-d application of this more general definition (all the values y can take on for some input value x).

Answer 10

The range is going to be the whole vector space, since every vector in the space can be represented as 3 times some other vector.

Answer 11

How so? can you give me a counter example?

Answer 12

if V is a vector space and v is an element of V,
then i thought cv is an element of V

Answer 13

where c is a number

Answer 14

im pretty sure vector spaces are closed under scalar multiplication, check out b) on this list of what conditions make a vector space:
http://tutorial.math.lamar.edu/Classes/LinAlg/VectorSpaces.aspx

Answer 15

Never mind, you're right - I was thinking of generalized vector sets.

Answer 16

so its the whole vector space that is the range

Answer 17

Yes

Answer 18

(I need to stop doing math on no sleep)

Answer 19

me too
i haven't slept

Answer 20

my hours are totally messed up

Answer 21

Another way to know the the range is the whole vector space is that the dim(Null Space)+ dim(range) = the dim(V), and since the dim(Null Space) = 0, the dim of the range must be all of V

Answer 22

yeah, this website totally ruined my sleeping schedule. >.>

Answer 23

why is this addictive?
its not even a drug

Answer 24

Well, I'm a mathematician by trade, so I'm already hooked. (Which makes stupid mistakes all the more embarrassing.)

Answer 25

Nothing wrong with mistakes. we all make them. being able to admit ones own mistakes shows great character.

Answer 26

That's something I truly love about math - there's usually a right answer. I'm not always the one who has it, but if I do I can prove it and if I don't it can be proven to me.