SOLUTION: Hi, I'm having some trouble with the concept of linear operators. I think of them as functions on sets, but still can't imagine actual examples. Given the properties of linear oper

Question 12549: Hi, I'm having some trouble with the concept of linear operators. I think of them as functions on sets, but still can't imagine actual examples. Given the properties of linear operators, these should be simple ones (though I still haven't come up with answers), so I'm hoping you can give me some ideas to help me grasp the linear operator a bit better. Here we go! 1) I'm looking for a linear operator from V->V such that T^2=0 but T does not =0. Then, 2) Given two linear operators (say, T, U) from V-> V, I'm looking for TU=0 but UT does not =0. The generality of T and U just really throws me off for some reason. Thanks again for your time. I really appreciate it.
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Linear operators are not merely functions on sets,they are linear transfomrations(preserving addition and scalar multiplication of vectors on vector space.
I feel very strange that you haven't mentioned vector space, or matrices.
Without them, how can you start solving the questions ?
1) I'm looking for a linear operator from V->V such that T^2=0 but T does not =0.
Sol: Define --> generated by
T(1,0) = (0,1) and T(0,1) = (0,0)
[Note : (1,0) & (0,1) are standard basis of unitcolumn vectors]
Let B = {(1,0),(0,1)}
The matrix A = [T}B of T associated with the basis B as
[0 0]
[1 0]
or equivalently T(X) = AX for all column vector X in
Clearly,you can see that but A is not 0.
More precisely, T(a, b) = aT(1,0) + bT(0,1) = a(0,1) = (0,a)
for all (a,b) in
Check: Clearly,T is not the zero operator (why?) and
we see that (a,b) = T(0,a) = 0*T(1,0) = (0,0) for all real a,b
2) Given two linear operators (say, T, U) from V-> V, I'm looking for TU=0 but UT does not =0
Sol: Similarly to the example in 1)
Let T ,U be two linear operators in defined by
T = (as matrix A)
[0 1]
[0 0] and
U = (as matrix B)
[1 0]
[0 0] then we have
TU =
[0 0]
[0 0] but UT =
[0 1]
[0 0] (not zero operator)
More precisely, define --> by T(X) = AX
and --> by T(X) = BX where X is any column vector of .
AX & BX are products of matrices.
Make sure you do understand the examples above and work hard.
Kenny

RELATED QUESTIONS
Hi, I am having some trouble with a Geometry question for an investigation, which... (answered by stanbon,xinxin)

Hi, I am having some trouble with some homework questions. I need to display the... (answered by Alan3354)

I'm still having trouble with the graph method.Could someone please show me how to do a... (answered by jim_thompson5910)

I'm having some trouble with this : What is the equation of a parabola with vertex (0,0) (answered by Fombitz)

A boat travels 5 mph in still water. In the time it takes to travel 42 miles downstream,... (answered by stanbon)

Hi there, I am having trouble with finding an equation for two sets of coordinates. for (answered by lwsshak3)

Hi, I am having trouble with a problem on my homework, I hope some one can help me,The... (answered by frostusna,checkley75)

Hi, I am having trouble with this question: Find all possible sets of 4 consecutive... (answered by richard1234)

Hi, my name is Grace Rogers and I am having trouble with some rational functions. I get... (answered by Fombitz)