The charger for a cellphone contains a transformer that reduces 120 V AC to 4.92 V AC to charge the 3.7 V battery. Suppose the secondary coil contains 50 turns and the charger supplies 680 mA. Calculate (a) the number of turns in the primary coil, (b) the current in the primary, (c) the power transformed and (d) repeat parts (a) to (c) with a 240 V AC to 6.78 V AC

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective.

Note: Don’t forget that bijective maps are precisely those that have an inverse!

Consider an unweighted, undirected graph G = <V, E>. The neighbourhood of a node v ∈ V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v).

(a) Design an algorithm that returns a list containing the neighbourhood degree for each node v ∈ V, assuming that the input is the graph, G, represented using an adjacency list. Each item i in the list that you generate will correspond to the correct value for the neighbourhood degree of node vi. Your algorithm should be presented in unambiguous pseudocode. Your algorithm should have a time complexity value O(V +E).

(b) If an adjacency matrix was used to represent the graph instead of an adjacency list, what is the new value for the time complexity? Justify your answer by explicitly referring to the changes that would be necessary to your algorithm from part (a).

1. For a map f : V ?? W between vector spaces V and W to be a linear map it must preserve the structure of V . What must one verify to verify whether or not a map is linear?

2. For a map f : V ?? W between vector spaces to be an isomorphism it must be a linear map and also have two further properties. What are those two properties? As well as giving the names of the properties, explain what the names mean.

3.Every linear transformation is an isomorphism, but the isomorphism f : x y ?? x y is not a linear transformation. Why

suppose that T : V → V is a linear map on a finite-dimensional vector space V such that dim range T = dim range T2. Show that V = range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range T = {0}, and apply the fundamental theorem of linear maps.)

Suppose V and V0 are finitely-generated vector spaces and T : V → V0 is a linear transformation with ker(T) = {~ 0}. Is it possible that dim(V ) > dim(V0)? If so, provide a specific example showing this can occur. Otherwise, provide a general proof showing that we must have dim(V ) ≤ dim(V0).

What is the concentration in % (v/v) of a methanol solution prepared by mixing 35.0 mL of methanol with 700 mL of distilled water?

Let T: V →W be a linear transformation from V to W.

a) show that if T is injective and S is a linearly independent set of vectors in V, then T（S） is linearly independent.

b) Show that if T is surjective and S spans V,then T（S） spans W.

Please do clear handwriting!

The Voltage amplifiers are available with Avoc = 8 V / V,
Rin = 1.8 kΩ, Ro = 850 Ω. With a 12V DC power source, each amplifier
consumes 1.5 mA average current.

a. How many amps do you need to cascade to get so
minus a voltage gain of 1000 with a load resistance of 1.0 kΩ?
b. What is the voltage gain Av obtained? (Answer with an integer number
rounded)
c. For the cascade connection, find the open circuit voltage gain.
(Answer with a rounded integer)
d. If you have a 1.5 mV input, how efficient is the equivalent amplifier?
and. 
e.Find the transconductance of the complete circuit

Given two graphs G = [V ; E] and G0 = [V 0 ; E0 ], and an isomorphism, f : V → V 0 , and making direct use of the formal definition for isomorphism:

(a) Explain why G and G0 must have the same number of vertices.

(b) Explain why G and G0 must have the same number of edges.

(c) Explain why G and G0 must have the same degree sequences.

(d) Given two vertices, u, v ∈ V explain why: u is connected to v → f(u) is connected to f(v)

Note: Problems that ask you to “explain” are asking for responses that can be less formal than problems that ask you to “prove”. Nonetheless, responses need to be sufficiently precise and based on definitions and theorems as given in the text. Explanations should be concise, but care must be taken to ensure that explanations are at an appropriate level of detail and will be clear to the intended reader.