Question

11.43. Let 1 ≤ *m* ≤ *n*. Show that

(a). *K _{m}* ∩

(b). *K _{m}* ∪

(c). *K _{m}* ⊆

Answer #1

Let1≤ m ≤ n. Show that
(a). Km ∩ Kn = Km,
(b). Km ∪ Kn = Kn,
(c). Km ⊆ Kn.

Find the diameters of Kn (Connected graph with n vertices), Km,n
(Bipartite graph with m and n vertices), and Cn (Cycle graph with n
vertices). For each, clearly explain your reasoning.

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 a, b, and n be integers with n > 1 and (a, n) = d.
Then
(i)First prove that the equation a·x=b has solutions in n if and
only if d|b.
(ii) Next, prove that each of u, u+n′, u+ 2n′, . . . , u+
(d−1)n′ is a solution. Here,u is any particular solution guaranteed
by (i), and n′=n/d.
(iii) Show that the solutions listed above are distinct.
(iv) Let v be any solution. Prove that v=u+kn′ for...

. Let n ∈ N. Prove (by induction) that n =
2knmn for some nonnegative kn ∈ Z
and some odd mn ∈ N. (Again, kn and mn may
depend on n.)

Consider the complete bipartite graph Kn,n with 2n vertices. Let
kn be the number of edges in Kn,n. Draw K1,1, K2,2 and K3,3 and
determine k1, k2, k3. Give a recurrence relation for kn and solve
it using an initial value.

Let
m,n be integers. show that the intersection of the ring generated
by n and the ring generated by m is the ring generated by their
least common multiple.

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

Let A be an n × n-matrix. Show that there exist B, C such that B
is symmetric, C is skew-symmetric, and A = B + C. (Recall: C is
called skew-symmetric if C + C^T = 0.) Remark: Someone answered
this question but I don't know if it's right so please don't copy
his solution

Let A be an m × n matrix, and Q be an n × n invertible
matrix.
(1) Show that R(A) = R(AQ), and use this result to show that
rank(AQ) = rank(A);
(2) Show that rank(AQ) = rank(A).

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