Given two sets S and T, the direct product of S and T is the set of ordered pairs S × T = {(s, t)|s ∈ S, t ∈ T}.Let V, W be two vector spaces over F.

(a) Prove that V × W is a vector space over F under componentwise addition and scalar multiplication (i.e. if (v1, w1),(v2, w2) ∈ V × W, then (v1, w1) + (v2, w2) = (v1+w1, v2+w2) and a(v, w) = (av, aw) for any (v, w) ∈ V ×W, a ∈ F).

(b) If dim V = n and dim W = m, prove that dim V × W = n + m by constructing a basis.

1.) Translate the following C code to MIPS assembly code. Assume that the variables f, g, h, i, and j are assigned to registers $s0, $s1, $s2, $s3, and $s4, respectively. Assume that the base address of the arrays A and B are in registers $s6 and $s7, respectively

  B[8] = A[i-j];

2.Translate the following C code to MIPS assembly code. Assume that the values of v in $a0, k in $a1, temp in $t0.

// leaf procedure that swaps v[k] and v[k+1]

void swap(int v[], int k)

{
int temp;
temp = v[k];
v[k] = v[k+1];
v[k+1] = temp;
}

1) Of four properties that define the conditions of gas (pressure, Volume, temperature and moles), which two are studied in the Boyle's Law experiment? explain briefly how the remaining two are unchanged during the experiment.

2) Which two are studied in Part B of this experiment? Explain briefly how the remaining two are unchanged during the experiment.

3) when a flask of gas is cooled, the pressure of the gas will change. what would happen to the gas if it were cooled to an extremely low temperature?

2
Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and w=(3,5,4,4).

2.1
Construct a basis for the vector space spanned by u, v and w.

2.2
Show that c=(1,3,2,1) is not in the vector space spanned by the above vectors u,v and w.

2.3
Show that d=(4,9,17,-11) is in the vector space spanned by the above vectors u,v and w, by expressing d as a linear combination of u,v and w.

Find an article relating to Time Value of Money or Asset Valuation techniques from any medium, briefly summarize the article, explain how you found the article particularly useful or timely, and give your personal reactions to the article. Be sure to provide the title of the article, the author(s), and the reference for the article.

Find an article relating to Time Value of Money or Asset Valuation techniques from any medium, briefly summarize the article, explain how you found the article particularly useful or timely, and give your personal reactions to the article. Be sure to provide the title of the article, the author(s), and the reference for the article.

1) Book Publishing

You are the owner of a publishing firm and you have a new author that you plan to publish. It is an action/espionage novel. You believe that the author has a good book, but it is her first book and you don’t really know what the sales numbers will look like. As such, you want to do a break even analysis to find out how many books you have to sell in order to get back your initial investment.

The book will be published in paperback sized 6”x9”. The initial set up cost for setting up the press for the book is $900 dollars. After the additional cost for setting up the press, each book will cost $2.19 each to make.

a) How much will it cost to print _ books?

o 100 books

o 200 books

o 500 books

o 1000 books

o 2000 books

b) What is the cost per book for each quantity?