Question

Let N be a nilpotent mapping V and letγ:V→V be an isomorphism. 1.Show that N and γ◦N◦γ−1 have the same canonical form 2. If M is another nilpotent mapping of V such that N and M have the same canonical form, show that there is an isomorphism γ such that γ◦N◦γ−1=M

Answer #1

Suppose S and T are endomorphisms of V and γ:V→W is an
isomorphism such thatγ◦T◦γ−1 and γ◦S◦γ−1 have the same kernel. Show
that S and T have the same kernel

Let
R be a ring, and let N be an ideal of R.
Let γ : R → R/N be the canonical homomorphism.
(a) Let I be an ideal of R such that I ⊇ N.
Prove that γ−1[γ[I]] = I.
(b) Prove that mapping
{ideals I of R such that I ⊇ N} −→ {ideals of R/N} is a
well-defined bijection between two sets

(Linear Algebra)
A n×n-matrix is nilpotent if there is a "r" such that
Ar is the nulmatrix.
1. show an example of a non-trivial, nilpotent 2×2-matrix
2.let A be an invertible n×n-matrix. show that A is not
nilpotent.

In R^2, let u = (1,-1) and v = (1,2).
a) Show that (u,v) form a basis. Call it B.
b) If we call x the coordinates along the canonical basis and y
the coordinates along the ordered B basis, find the matrix A such
that y = Ax.

Show that if G is a nilpotent group and 1 ≠ N, and N is normal
in G, then N ∩ Z(G) ≠ 1.

1.a Implication: If an endomorphism f : V → V is not an
isomorphism, then it must have a non-trivial kernel. Use this
implication, together with the definition of determinant given in
class, to show that if for an endomorphism f : V → V we have det(f)
6 /= 0, then in fact f is an isomorphism.
1.b.
The Rank-Nullity Theorem applies to a linear map f : V → W
(where V is finite dimensional) and claims that:...

Let n be in N and let K be a field. Show that for a linear map T
: Kn to Kn the following statements are
equivalent:
1. The map T is one-to-one (injective).
2. The map T is onto (surjective).
3. The map T is invertible.
4. The map T is an isomorphism.

Let V be a vector subspace of R^n for some n?N. Show that if
k>dim(V) then the set of any k vectors in V is dependent.

Let G = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.

Let V be a vector space with dimV = n.
Show that : Any spanning set for V consisting of exactly n
vectors is a basis for V.

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