Hi I know how to do problems most problems on Kernel and Range but this one for some reason I can't do

Please guide me towards how to do this one

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- Dec 16th 2012, 07:30 PMohYeahHint on how to approach this Kernal and Range problem
Hi I know how to do problems most problems on Kernel and Range but this one for some reason I can't do

Please guide me towards how to do this one

Attachment 26268 - Dec 16th 2012, 07:46 PMDevenoRe: Hint on how to approach this Kernal and Range problem
suppose p(x) is in ker(T). what does this mean? it means T(p(x)) = 0 (the polynomial, not the number).

since T(p(x)) = x^{2}p(x), we have 2 possibilities:

a)x^{2}is the 0-polynomial. is this ever true?

b) p(x) is the 0-polynomial. what does this say about dim(ker(T))?

note that dim(P_{2}) = 3 (the "2" is misleading). what does the rank-nullity theorem tell you about dim(im(T))? - Dec 16th 2012, 08:12 PMohYeahRe: Hint on how to approach this Kernal and Range problem
a) x^2 would equal zero only if x=0

b) if p(x) is the zero polynomial then wouldn't that mean that dim(ker(T)) = 0

i don't understand what you mean by im(T)

The R_N says dim(range) + dim(kernel) = dim(domain)

I hope I am saying anything right - Dec 16th 2012, 08:34 PMohYeahRe: Hint on how to approach this Kernal and Range problem
I am sorry if i am saying something really stupid and it might make me look like an idiot, but I am just trying to read this lesson now and do these questions on the study guild.

so please excuse anything out of this world i might say - Dec 16th 2012, 08:37 PMDevenoRe: Hint on how to approach this Kernal and Range problem
im(T) is the image, or range, of T. it's just an abbreviation (im(T) looks cooler than rang(T)).

your conclusion for (a) is incorrect. do not confuse the expression x^{2}, where x is a real number, with the POLYNOMIAL x^{2}(which is a FUNCTION, not a number). - Dec 16th 2012, 08:54 PMohYeahRe: Hint on how to approach this Kernal and Range problem
oh yeah I knew that's where i probably went wrong, so then for (a) x^2 can't ever be 0 polynomial

so then are we only dealing with part (b) here when it comes to the kernel so far it seams the only thing in the kernel is the {0}, so its dim is 0.

so that would leave the range to be dim(domain) -dim(kernel). so its 3 - Dec 16th 2012, 09:02 PMDevenoRe: Hint on how to approach this Kernal and Range problem
can you think of a 3-dimensional subspace of P

_{4}whose elements all contain an x^{2}term?

(hint: perhaps {x^{2},x^{3},x^{4}} might form a basis?) - Dec 16th 2012, 09:06 PMohYeahRe: Hint on how to approach this Kernal and Range problem
I am really sorry I don't know if its me or the time since its 1 AM but I am lost :(

- Dec 16th 2012, 09:10 PMDevenoRe: Hint on how to approach this Kernal and Range problem
do you know any bases at all for polynomial spaces?

- Dec 16th 2012, 09:20 PMohYeahRe: Hint on how to approach this Kernal and Range problem
well when I was given a polynomial like say (3x^2+2x+5) I would look at it in terms of R^3 (3,2,5) and from there i would work with questions regarding polynomials so i never really had to know more then that till now i guess

can you please elaborate on what i should know, I know I am taking alot of your time, and i really appreciate your efforts so far - Dec 16th 2012, 09:36 PMDevenoRe: Hint on how to approach this Kernal and Range problem
ok, that's good. now if you're identifying 3x

^{2}+2x+5 with (3,2,5) what is your basis?

that is you have (fill in the blanks): 3___ + 2____ + 5____ ? the stuff in the blanks are your basis vectors. - Dec 16th 2012, 09:50 PMohYeahRe: Hint on how to approach this Kernal and Range problem
oh so my x^2, x,1 are my bases vectors.

i see

so to go back to what we were doing you state " perhaps {x2,x3,x4} might form a basis?" if its for P4 or in R^5 (0,0,1,1,1)

would that be enough to form a bases, since what i know is that a bases would have to span and isn't it to small to span all of R^5 or P4

and i think i lost though of how we are relating this to the original question about Kernal and Range, since i still don't know the answer - Dec 16th 2012, 10:06 PMDevenoRe: Hint on how to approach this Kernal and Range problem
things you should know (if you don't, re-read your texts)

1. a subset of a linearly independent set is linearly independent

corollary: a subset of a basis is linearly independent

2. the span of a linearly independent set is a subspace for which the LI set forms a basis

corollary: a subset of a basis determines a subspace

P_{4}and R^{5}have the same dimension so they "act alike" (as vector spaces). but they are not "the same" there's just a CORRESPONDENCE between the two:

ax^{4}+bx^{3}+cx^{2}+dx+e <--> (a,b,c,d,e)

inducing the following correspondence between the two bases:

x^{4}<--> (1,0,0,0,0)

x^{3}<--> (0,1,0,0,0)

x^{2}<--> (0,0,1,0,0)

x <--> (0,0,0,1,0)

1 <--> (0,0,0,0,1)

*********

in terms of your linear mapping for your problem, the range of T is {f(x) in P_{4}: f(x) = ax^{4}+bx^{3}+cx^{2}}

that is f(x) = x^{2}(p(x)) , where p(x) = ax^{2}+bx+c.

if you want to think of this as the subspace of R^{5}spanned by (1,0,0,0,0) ("x^{4}"), (0,1,0,0,0) ("x^{3}") and (0,0,1,0,0) ("x^{2}") that's "sort of" ok, just realize you're using an EQUIVALENCE, not an EQUALITY.