Invertible, Kernal and Range of a Linear Transformation

[hkus10]

<font face="Arial">1) Let L:R3 >>>R3 be defined by<br>

L([1 0 0]) = [1 2 3],<br>

L([0 1 0]) = [0 1 1],<br>

L([0 0 1]) = [1 1 0]<br><br>

How to prove that L is invertible? I have the idea of one-to-one and onto, but I do not know how to apply them to this proof.<br><br>

2) Find a linear transformation L:R2 >>>R3 such that {[1 -1 2],

[3 1 -1]} is a basis for range (L). How can I approach this problem?<br><br>

3) Let S = {v1,…,vn} be an ordered basis for vector space V . Let<br>

L :V →R^n be given by L(v) = [v]S . Prove that L is an<br>

isomorphism( prove that the linear transformation is one-to-one and onto)<br>

What I know so far is that {[v1]s, [v2]s,...,[vn]s} is an ordered basis

for R^n and v can be written in a unique way sych that v = a1v1+...+anvn

= 0.<br>

How can I go from there?<br><br>

4) If L:V>>>W is a linear transformation of a vector space V

into a vector space W and dim V = dim W, then prove that if L is

one-to-one, then it is onto.<br>

I get a point that dim(W) = dim(range(L)). What I am trying to get to is

W = range(L). Then, by definition, L is onto. My question is that how

can I prove that dim(W) = dim(range(L)) >>> W = range(L) and

by what Thm of defin? If not, how should I approach this question?<br><br>

5) Let L:R^n>>>R^m be a linear transformation defined by L(x) =

Ax, for x in R^n. Then, L is onto if and only if rank A = m.<br>

In this question, is the nullity of A equal to the nullity of L?<br><br>

Thanks

</font>

Edited April 24, 2011 by hkus10

[keelanz]

and what homework is this for?

 

ive never seen any sort of language like this, i dont get the syntax sorry.

 

what level of language is this? im actually very interested =D

 

 

ahh i get whats happening, your declaring things and asking some sort of propositional question in relation to certain areas of computing?

 

i think theres some sort of mix up though html seems all over the place so it doesnt have any sort of logic that makes sense without taking out html?

 

i dont really know any of the concepts at work here so ill let someone take over from where i left off, this is all rather confusing me.

 

if you want to feed me more information ill have a go

Edited April 26, 2011 by keelanz

[A Tripolation]

ive never seen any sort of language like this' date=' i dont get the syntax sorry.

[/quote']

 

He copy-pasted from something that had straight HTML code in it, yes, but the questions he is asking has to do with Linear Algebra. And he just wants answers. Which is why I assume no one here has tried to help.

Edited April 27, 2011 by A Tripolation