(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2

(b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where we write V1+V2 to be the subspace of V spanned by V1 and V2 .

Prove the following:

Let V and W be vector spaces of equal (finite) dimension, and let T: V → W be linear. Then the following are equivalent.

(a) T is one-to-one.

(b) T is onto.

(c) Rank(T) = dim(V).

According to Boyle’s law, for a certain gas, pressure an volume are inversely proportional: P = 800/V where V is measured in and pressure P in pounds per square inch.
a) Find the rate of change of P when V=20 , and interpret its meaning (carefully state the units).
b) Find the average rate of change of P when V changes from 20 to 25 . Compare your result with a).

Two charged point-like objects are located on the x-axis. The point-like object with charge

q₁ = 5.80 µC

is located at

x₁ = 1.25 cm

and the point-like object with charge

q₂ = −2.18 µC

is located at

x₂ = −1.80 cm.

(a)

Determine the total electric potential (in V) at the origin.

V

(b)

Determine the total electric potential (in V) at the point with coordinates (0, 1.50 cm).

V

Let K_n denote the simple graph on n vertices.
(a) Let v be some vertex of K_n and consider K _{n − v}, the graph obtained by deleting
v. Prove that K _{n − v} is isomorphic to K n−1 .
(b) Use mathematical induction on n to prove the following statement:
K _n , the complete graph on n vertices, has n(n-1)/2
edges

Let W denote the set of English words. For u, v ∈ W, declare u ∼ v provided that u, v have the same length and u, v have the same first letter and u, v have the same last letter.

a) Prove that ∼ is an equivalence relation.

b) List all elements of the equivalence class [a]

c) List all elements of [ox]

d) List all elements of [are]

e) List all elements of [five]. Can you find more than 15?

f) Bonus. Find all three letter words x such that [x] has 5 elements.

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V₁ and V₂, such that every edge joins a vertex in V₁ with a vertex in V₂. This means no edges are within V₁ or V₂ (or symbolically: ∀u, v ∈ V₁, {u,v} ∉ E and ∀u,v ∈ V₂, {u,v} ∉ E).

8(a) Show that the complete graph K₂ is a bipartite graph.

8(b) Prove that no complete graph K_n, where n > 2, is a bipartite graph.

8(c) Prove that every rooted tree forms a bipartite graph.

1.-

A) A particle of total energy 9V₀ is incident from the -x-axis on a potential given by V(x) = 8V₀ for x < 0, V(x) = 0 for 0 < x < a and V(x) = 5V₀ for x > a. Find the probability that the particle will be transmitted on through to the positive side of the x-axis, x > a.

B) Consider a particle incident from the left on a potential step which is defined as V(x) = V₁ for x < 0 and V(x) = V₂ for x > 0 with V₁ < V₂. Find the solution of the Schrödinger equation for the following cases: (i) V₁ < E < V₂ (ii) E > V₂.