Questions about Linear Algebra

[Artusartos]

Homework Statement

Let $A \in M_n(F)$ and $v \in F^b$.

Also...$[g \in F[x] : g(A)(v)=0] = Ann_A (v)$ is an ideal in F[x], called the annihilator of v with respect to A. We know that $g \in Ann_A(v)$ if and only if f|g in F[x]. Let $V = Span(v, Av, A^2v, ... , A^{k-1}v).$. V is teh smallest A-invariant subspace containing v. We denote the fact by writing V=F[x]v. This corresponds to the F[x]-module structure on $F^n$ induced by multiplication by A. We also know that $v, Av, A^2v, ... , A^{k-1}v$ is a basis, B, of v.

Now these are the questions...

1) Define $Ann_A(V) =[g \in F[x] : g(A)(w) = 0 for all w \in V].$ Show that $Ann_A(V)=Ann_A(v)$

2) Let T: V -> V be induced by multiplication by A: T(w)=Aw for $w \in V$. Show that $Ann_A(V) = [g \in F[x] : g(T) = 0] = [g \in F[x]: g([T]_B)=0]$.

Here the first one means that g(T): V -> V is the 0-transformation and the second one means that g([T]_B) is the 0-matrix. Since $Ann_A(V) = (f) = [g \in F[x]: f|g]$, we write $f=min_T(x)$, the monic polynomial of lowest degree with f(T)=0.

Homework Equations

The Attempt at a Solution

My answers:

1) In order to show that $Ann_A(V)=Ann_A(v)$, I need to show that $Ann_A(V) \subset Ann_A(v)$ and $Ann_A(v ) \subset Ann_A(V)$. It is clear that $Ann_A(v ) \subset Ann_A(V)$, since v is in V.

In order to show that $Ann_A(V) \subset Ann_A(v)$...

we say that, since $Ann_A(V) = [g \in F[x] : g(A)(w) = 0 for all w \in V]$, then g(A) must be zero for A restricted to V. So, assuming that it is zero when A is restricted to V, $Ann_A(V) \subset Ann_A(v)$. So they must be equal.

2) We say g(T) = g(A), where A is restricted to v.

$g(x) = c_0 + c_1x +... + c_tx^t$ and $g(T)w= (c_0 + c_1T + ... + c_tT^t)(w) = c_0w + c_1Tw +... + c_t(T)^t)(w) = c_0w + c_1Aw + ... + c_t(A)^tw = (c_0 + c_1A + ... + c_tA^t)(w) = g(A)w$.

So $[g \in F[x]: g(A).w = 0 for all w \in V]$ = $[g \in F[x]: g(T).w=0 for all w \in V]$ = $[g \in F: g(T) = 0]$.

Now for the second one, we do the exact same thing, except that we say $g([T]_B) = g(A)$, where g(A) is restricted to $[v]_B$.

So...

$g(x) = c_0 + c_1x +... + c_tx^t$ and $g([T]_B)w= (c_0 + c_1[T]_B + ... + c_t{[T]_B}^t)(w) = c_0w + c_1[T]_Bw +... + c_t{[T]_B}^t)(w) = c_0w + c_1Aw + ... + c_t(A)^tw = (c_0 + c_1A + ... + c_tA^t)(w) = g(A)w$.

So $[g \in F[x]: g(A).w = 0 for all w \in V]$ = $[g \in F[x]: g([T]_B).w=0 for all w \in V]$ = $[g \in F: g([T]_B) = 0]$

Do you think my answers are correct? If not, then can you tell me why?

Thanks in advance

 

[mighty2000]

!

I would like to provide some additional explanation and clarification to your answers.

1) Your explanation for why Ann_A(V) \subset Ann_A(v) is correct. However, you need to also show that Ann_A(v) \subset Ann_A(V) in order to prove that they are equal. To do this, you can use the fact that V is the smallest A-invariant subspace containing v, and therefore any polynomial that annihilates all elements of V must also annihilate v.

2) Your reasoning for showing Ann_A(V) = [g \in F[x] : g(T) = 0] and Ann_A(V) = [g \in F[x] : g([T]_B) = 0] is correct. However, you need to show that these two sets are equal, not just that they both equal Ann_A(V). This can be done by showing that any polynomial that annihilates T also annihilates [T]_B, and vice versa.

Additionally, you may want to clarify what you mean by "restricted to v" and "[v]_B". It would also be helpful to explain the significance of the monic polynomial of lowest degree with f(T)=0 in relation to Ann_A(V).

Overall, your answers show a good understanding of the concepts involved, but could benefit from some additional clarification and explanation.