I have a couple of questions on these topics that i've been trying for hours....Any help would be very much appreciated!!
1) Find bases for both N(T) & R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Use the appropriate theorems to determine whether T is 1-1 or onto..
T: R[sup:2k94fxbt]2[/sup:2k94fxbt] -> R[sup:2k94fxbt]3[/sup:2k94fxbt] defined by T(a[sub:2k94fxbt]1[/sub:2k94fxbt], a[sub:2k94fxbt]2[/sub:2k94fxbt]) = (a[sub:2k94fxbt]1[/sub:2k94fxbt]+ a[sub:2k94fxbt]2[/sub:2k94fxbt], 0, 2a[sub:2k94fxbt]1[/sub:2k94fxbt] - a[sub:2k94fxbt]2[/sub:2k94fxbt])
I'm not too sure where to go from finding the N(T)...I think I have a clue about the R(T)...but not totally sure...
All I got so far is that N(T) : a[sub:2k94fxbt]1[/sub:2k94fxbt] + a[sub:2k94fxbt]2[/sub:2k94fxbt] = 0
2a[sub:2k94fxbt]1[/sub:2k94fxbt] - a[sub:2k94fxbt]2[/sub:2k94fxbt] = 0 which gives a matrix...not sure where to go from there...
2) Let V and W be vector spaces and T: V --> W be linear.
a) Prove that T is 1-1 if and only if T carries linearly independent subsets of V onto linearly independent subsets of W.
b) Suppose that T is 1-1 and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent.
c) Suppose B {v[sub:2k94fxbt]1[/sub:2k94fxbt], v[sub:2k94fxbt]2[/sub:2k94fxbt],...,v[sub:2k94fxbt]n[/sub:2k94fxbt]} is a basis for V and T is 1-1 and onto. Prove that T(B) = {T(v[sub:2k94fxbt]1[/sub:2k94fxbt]), T(v[sub:2k94fxbt]2[/sub:2k94fxbt]),...,T(v[sub:2k94fxbt]n[/sub:2k94fxbt])} is a basis for W
1) Find bases for both N(T) & R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Use the appropriate theorems to determine whether T is 1-1 or onto..
T: R[sup:2k94fxbt]2[/sup:2k94fxbt] -> R[sup:2k94fxbt]3[/sup:2k94fxbt] defined by T(a[sub:2k94fxbt]1[/sub:2k94fxbt], a[sub:2k94fxbt]2[/sub:2k94fxbt]) = (a[sub:2k94fxbt]1[/sub:2k94fxbt]+ a[sub:2k94fxbt]2[/sub:2k94fxbt], 0, 2a[sub:2k94fxbt]1[/sub:2k94fxbt] - a[sub:2k94fxbt]2[/sub:2k94fxbt])
I'm not too sure where to go from finding the N(T)...I think I have a clue about the R(T)...but not totally sure...
All I got so far is that N(T) : a[sub:2k94fxbt]1[/sub:2k94fxbt] + a[sub:2k94fxbt]2[/sub:2k94fxbt] = 0
2a[sub:2k94fxbt]1[/sub:2k94fxbt] - a[sub:2k94fxbt]2[/sub:2k94fxbt] = 0 which gives a matrix...not sure where to go from there...
2) Let V and W be vector spaces and T: V --> W be linear.
a) Prove that T is 1-1 if and only if T carries linearly independent subsets of V onto linearly independent subsets of W.
b) Suppose that T is 1-1 and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent.
c) Suppose B {v[sub:2k94fxbt]1[/sub:2k94fxbt], v[sub:2k94fxbt]2[/sub:2k94fxbt],...,v[sub:2k94fxbt]n[/sub:2k94fxbt]} is a basis for V and T is 1-1 and onto. Prove that T(B) = {T(v[sub:2k94fxbt]1[/sub:2k94fxbt]), T(v[sub:2k94fxbt]2[/sub:2k94fxbt]),...,T(v[sub:2k94fxbt]n[/sub:2k94fxbt])} is a basis for W