Linear Algebra 2270
Homework 6
preparation for the quiz on 07/01/2015
Problems:
1. Determine whether the following are linear transformations. Prove your claim.
x
(a) T ∶ R → R2 , T (x) = [
]
x+1
(b) T ∶ R → R2 , T (x) = [
x
]
4x
u
u + u22
(c) T ∶ R2 → R2 , T ([ 1 ]) = [ 1
]
u2
u2 + π
⎡ u1 + 2u2 ⎤
⎥
⎢
u1
⎥
⎢
2
3
(d) T ∶ R → R , T ([ ]) = ⎢−2u1 − 4u2 ⎥
⎥
⎢
u2
⎥
⎢
u2
⎦
⎣
2
R
(e) T ∶ R → R
u
∀x∈R T ([ 1 ]) (x) = (u1 − u2 ) sin(x) + u2 cos(x)
u2
2. Find the kernel and range of the linear transformations of the previous problem.
3. Let T ∶ V → W be a linear transformation. Prove that the range of T is a subspace of W .
(Hint): Notice that w ∈ range(T ) ⇐⇒ ∃v∈V T (v) = w
4. The order of transformations matters. If you are making a pie, the following are not equivalent:
(a) Mix the ingredients, then bake.
(b) Bake the ingredients, then mix them.
Or if you are changing oil in a car the following are not equivalent
(a) Drain the oil, then add new oil.
(b) Add new oil, then drain the oil.
More mathematically, let f, g be two functions
f ∶ Af → Bf , g ∶ Ag → Bg
Then f ○ g does not have to be the same as g ○ f .
Notice that to consider f ○ g one needs
B g = Af
(1)
B f = Ag
(2)
and then, f ○ g ∶ Ag → Bf .
To consider g ○ f , one needs:
and then, g ○ f ∶ Af → Bg .
But even if Af = Bf = Ag = Bg it might be that f ○ g ≠ g ○ f .
Find an example of two functions f, g ∈ RR for which f ○ g ≠ g ○ f .
Find an example of two linear transformations f, g ∶ R2 → R2 such that f ○ g ≠ g ○ f ?
5. (extra problem, will not be on the quiz) Find the kernel and range of the derivative. More precisely,
find the kernel and the range of T :
Let C(R) denote a set of continuous functions f ∶ R → R
Let C 1 (R) denote a set of functions f ∶ R → R that have a continuous derivative.
T ∶ C 1 (R) → C(R), T (f ) = f ′ .
1