Proving Ker(f) is a Subset of Ker(h) in Linear Algebra Problem 1

[acm]

1. The maps F:V -> V and G:V -> V are both linear, where V is a vector space. Suppose that the map H:V -> V is determined by h(v) = g(f(v)). Show that Ker(f) is a subset of Ker(h).

In the mapping F: V -> V, Ker(F) = (v element of V : F(v) = 0)
In the mapping G: V -> V, Ker(G) = (v element of V : G(v) = 0)

As the Kernel is the subset of the transformation then Ker(F) is a subset of V. I cannot see how I can proove that the vector space V is equal to the kernel of the transformation h.

 

[kesh]

1. The maps F:V -> V and G:V -> V are both linear, where V is a vector space. Suppose that the map H:V -> V is determined by h(v) = g(f(v)). Show that Ker(f) is a subset of Ker(h).

In the mapping F: V -> V, Ker(F) = (v element of V : F(v) = 0)
In the mapping G: V -> V, Ker(G) = (v element of V : G(v) = 0)

As the Kernel is the subset of the transformation then Ker(F) is a subset of V. I cannot see how I can proove that the vector space V is equal to the kernel of the transformation h.

the kernal isn't a subset of the transformation, it's a subset of the domain of the transformation.

also the question hasn't asked you to prove that V = ker(H)



to prove ker(F) is a subset of ker(H) show that v, element of ker(F), is an element of ker(H)

1. F(v) = 0, by v an element of ker(F)
   
2. G(F(v)) = 0 as F(v) = 0 from above and G(0) = 0 by properties of G being a linear transformation
   
3. so H(v) = 0, by defn H(v) = G(F(v)) and G(F(v)) = 0 above
   
4. so v is an element in ker(H)

 

[HallsofIvy]

1. The maps F:V -> V and G:V -> V are both linear, where V is a vector space. Suppose that the map H:V -> V is determined by h(v) = g(f(v)). Show that Ker(f) is a subset of Ker(h).

In the mapping F: V -> V, Ker(F) = (v element of V : F(v) = 0)
In the mapping G: V -> V, Ker(G) = (v element of V : G(v) = 0)

As the Kernel is the subset of the transformation then Ker(F) is a subset of V. I cannot see how I can proove that the vector space V is equal to the kernel of the transformation h.

"As the Kernel is a subset of the transformation"? What does this mean? A "transformation" is a function, not a set. Perhaps you meant to say "the kernel is a subset of the domain of the transformation". In this case the domain of F is V so Ker(F) is a subset of V. You don't want to prove "the vector space V is equal to the kernel of the transformation h" that's not said anywhere.

In general, to prove A is a subset of B, you start "if a is in A" and eventually prove "then a is in B". In this case A is Ker(f) and B is Ker(h). If a is in Ker(f) then f(a)= 0. What then is g(f(a))? What does that tell you about h(a)?