Let V -^Φ -> W be linear. Show that ker (Φ) is a subspace of V and Φ (V) is a subspace of W.

The fundamental (v = 0 ! v = 1) vibrational transition for H35Cl occurs at 2885 cm-1 . Close inspection of a high-resolution infrared spectrum of this transition shows the following absorptions (in cm-1 ) for the J ! (J-1) branch: 2865.10, 2843.62, 2821.56, 2798.94, 2775.76, 2752.04, 2727.78, 2703.01, and 2677.73. From these data, determine

a) the rotational-vibrational coupling constant, αe

b) the rotational constant, Be

c) the bond distance

In the Westway Coffees Corp. v. M. V. Netuno, why was the carrier liable for the error in the quantity of goods in the shipping container?

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!

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!