Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations V. Let u = (u1, u2) and v = (v1, v2).
Show that V is not a vector space.
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + 1 )
• ku = (ku1 + k − 1, ku2 + k − 1)
1)Show that the zero vector is 0 = (−1, −1).
2)Find the additive inverse −u for u = (u1, u2). Note: is not (−u1, −u2), so don’t write that.
3)Show that V is not a vector space.
In: Math
I have a collection of baseball cards, some from the 1984 season and some from other seasons. Considering only my valuable baseball cards, 60% of them are from 1984. Overall, 3% of my baseball cards are both valuable and from the 1984 season, and 38% of my cards are not valuable and not from the 1984 season. Fill out the table below
p(1984 and V) =
p(not 1984 and V) =
p(V) =
p(1984 and not V) =
p(not 1984 and not V) =
p(not V) =
p(1984) =
p(not 1984) =
In: Statistics and Probability
GVC_E(V,E)
1. C =Φ
2. while ( E≠ Φ)
3. do
4. { select an edge (u,v)∈E;
5. C = C∪{u}∪{v};
6. delete u and v from V and edges with u or v as an endpoint from E;
7. }
8. for each u∈C
9. { if C\{u} is a valid cover;
10. C = C\{u};
11. }
How can I complete the pseudo code on line 4 and line 8 by adding a heuristic specifying how to select a node
In: Computer Science
a.) Let T be a binary tree with n nodes. Define the lowest common ancestor (LCA) between two nodes v and w as the lowest node in T that has both v and w as descendants. Given two nodes v and w, write an efficient algorithm, LCA(v, w), for finding the LCA of v and w. Note: A node is a descendant of itself and v.depth gives a depth of a node v.
b.) What is the running time of your algorithm? Give the asymptotic tight bound (Q) of its running time in terms of n and justify your answer.
In: Computer Science
Loblaw Companies Limited (Loblaw) is a Canadian food retailer that owns 1,000 corporate and
franchise supermarkets that operate under 22 regional and market segment banners. Loblaw
brands include President’s Choice, No Name, Joe Fresh, T&T, Everyday Living, Exact,
Seaquest, Azami, and Teddy’s Choice. Loblaw is a public company, and its shares are listed on
the Toronto stock exchange.
Below you will find the 2016 and 2015 consolidated balance sheets (statement of financial
position). You will also find some excerpts from its notes section. All amounts are in millions of
Canadian dollars.
The fiscal year of the Company ends on the Saturday closest to December 31. Any references
below to 2015 relate to the fiscal year ended January 2, 2016, and any references below to 2016
relate to the fiscal year ended December 31, 2016.
REQUIRED:
a) Calculate the following ratios for both 2016 and 2015:
a. Current ratio
b. Quick ratio
b) Based on your calculation in a), comment on the liquidity of Loblaw and how/if
it has changed between fiscal year 2015 and 2016.
c) Cost of goods sold is $33,213 million for 2016 and $32,846 million for 2015.
Inventory balance was $4,309 million for 2014. Calculate the followings for both 2016
and 2015:
a. Inventory turnover
b. Days to sell inventory
d) Based on the brief description of Loblaw and your understanding of retail
business operations, do you think your calculations in c) are reasonable and why?
e) Use the information below from Note 12 “Inventories” and answer the
following:
a. Prepare a journal entry to record the write-down of inventories.
b. Give two examples and explain why a write-down of inventories is necessary for
Loblaw.
Note 12 Inventories
For inventories recorded as at December 31, 2016, the Company recorded $22 million as
an expense for the write-down of inventories below cost to net realizable value. The
write-down was included in cost of merchandise inventories sold.
f) Calculate the debt-to-equity ratio for 2016. Explain what this ratio measures
and why creditors want to see this ratio.
g) Answer the following questions:
a. Loblaw has unlimited number of authorized shares on each class of shares. Why
do many companies today prefer to have an unlimited authorized number of
shares?
b. List and explain three differences between common shares and preferred shares.
c. Based on the information available, are you able to determine the net income for
the year ended December 31, 2016? Show your detailed calculations or explain
why not.
d. The unit price for Loblaw’s common shares was $70.33 on December 31, 2016
and $63.92 on December 31, 2015. Note 24 “Share Capital” (not provided)
indicated the following:
2016 2015
Dividends declared per share ($):
Common Share $1.03 $0.0995
What is the dividend yield for common shareholders in each of 2016 and 2015? If
you are a common shareholder, are you happy to see the change and why? (3
marks)
Loblaw Companies Limited
Consolidated Balance Sheet
As of December 31
(in millions of Canadian Dollars)
2016 2015
Assets
Current Assets
Cash and cash equivalents $ 1,314 $ 1,018
Short term investments 241 64
Accounts receivable 4,048 4,115
Inventories 4,371 4,322
Prepaid expenses and other assets 230 336
Total Current Assets $ 10,204 $ 9,855
Non-current Assets
Fixed assets 11,592 11,558
Intangible assets 8,745 9,164
Goodwill 3,895 3,780
Total Non-current Assets $ 24,232 $ 24,502
Total Assets $ 34,436 $ 34,357
Liabilities
Current Liabilities
Bankindebtedness $ 115 $ 143
Trade payables 5,091 5,106
Provisionsand other liabilities 1,736 1,973
Total Current Liabilities $ 6,942 $ 7,222
Non-Current liabilities
Long term debt and other liabilities $ 14,466 $ 14,011
Total liabilities $ 21,408 $ 21,233
Equity
Share capital $ 7,913 $ 8,072
Retained Earnings 4,944 4,914
Contributed surplus 112 102
Accumulated other comprehensive
income 33 23
Non-controlling interest 26 13
Total Equity $ 13,028 $ 13,124
Total Liabilities and Equity $ 34,436 $ 34,357
In: Accounting
Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V = W = R^n to simplify things. Hint: show that the “preimage of” operation plays nice with set-complements, and then use the fact that every closed set is the complement of some open set. Note that R^n is both open and closed as a subset of itself.
In: Advanced Math
For each of the following statements, determine whether the statement is true or false. If you say the statement is true, explain why and if you say it is false, give an example to illustrate.
(a) If {u, v} is a linearly independent set in a vector space V, then the set {2u + 3v, u + v} is also a linear set independent of V.
(b) Let A and B be two square matrices of the same format. Then det (A + B) = det (A) + det (B).
(c) It is possible to find a non-zero square matrix A such that A^2 = 0.
(d) Let V be a vector space. If {v1, v2,. . . , vn} (with n ≥ 1) is a base of V and if {w1, w2,. . . , wm} is a generator system of V then n ≤ m.
In: Math
V and W are finite dimensional inner product spaces,T: V→W is a linear map
1A: Give an example of a map T from R2 to itself (with the usual inner product) such that〈Tv,v〉= 0 for every map.
1B: Suppose that V is a complex space. Show that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉
1C: Suppose T is a linear operator on a complex space such that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all v).
In: Advanced Math
Let V = R^2×2 be the vector space of 2-by-2 matrices with real
entries over
the scalar field R. We can define a function L on V by
L : V is sent to V
L = A maps to A^T ,
so that L is the “transpose operator.” The inner product of two
matrices B in R^n×n and C in R^n×n is usually defined to be
<B,C> := trace (BC^T) ,
and we will use this as our inner product on V . Thus when we talk
about
elements B,C in V being orthogonal, it means that <B,C> :=
trace (BC^T) = 0.
Problem 1.
1. First show that L is linear, so that L in B (V ).
2. Now choose a basis for the vector space V = R^2×2, and find the
matrix of
L with respect to your basis.
In: Advanced Math
In: Computer Science