General vector spaces Let V1,V2,V3 span a vector space V. Show that the Vector V1,V1+V2,V1+V2+V3 also span V. Solution Let V1 = p, V2=q, V3=r As they span, we may write ap+bq+cr = O where O is the zero element of V and the above relation holds only for a=b=c =0 for any a,b,c Now k(v1)+l(v1+v2)+m(v1+v2+v3)=O => (k+l+m)v1 +(l+m)v2+nv3=O => k+l+m=0 , l+m =0 , n=0 => k=l=m =0 => they are linearly independent with same dimension of the basis => new set of vectors also span V.

1. Gere Club Makers International manufactures golf clubs for amateur golfers and sells them in
complete sets. Each set of clubs sells for $250. The costs for the company for a typical year are
attached.
The company expects to sell 120,000 sets of clubs per year. The company has an average
income
tax rate of 25%.
REQUIRED: (1) How many sets of golf clubs must the company sell in order to break even?
Round your answer up to the next whole unit.
(2) How much sales in dollars must the company have in order to earn a before
tax profit of $150,000? Round your answer to nearest whole dollar.
(3) Suppose that the company desires to earn an after tax profit of $225,000.
How many sets of golf clubs must the company sell in order to achieve this
goal? Round your answer up to the next whole unit.
(4) Suppose that the company can eliminate the fixed manufacturing overhead
costs completely by increasing the variable manufacturing overhead cost to
$25 per unit. How many sets of golf clubs must the company now sell in
order to break even? Round your answer up to the next whole unit.
(5) Using the original cost information in the problem, assume that the company
intends to sell 150,000 sets of golf clubs for the current year. What price per
unit would it have to charge in order to earn a before tax profit of $300,000?
Round your answer to the nearest whole cent.
GERE CLUB MAKERS INTERNATIONAL
COSTS FOR THE COMPANY FOR A NORMAL YEAR
Direct Materials Cost Per Unit $ 100.00
Direct Labor Cost Per Unit 30.00
Variable Manufacturing Overhead Cost Per Unit 20.00
Fixed Manufacturing Overhead Cost Per Year 70,000
Variable Marketing and Admin. Cost Per Unit 10.00
Fixed Marketing and Admin. Cost Per Year 250,000
Solution
Req 1: Break even point: Fixed cost / CM per unit 320000 /90 = 3556 units Req
2: Desired profits: 150000 Desired contribution: 150000+320000 = 470000 Target
sales units: Desired cocntribution / CM per unit 470000/90 = 5223 units Req 3:

2. Desired profit after tax: 225000 Desired profit before tax: 225000/75% = 300,000
Desired contribution: 300,000+320000 =620000 Target sales units: Desired contribution/
CM per unit 620000/ 90 = 6889 units Req 4: Revised fixed cost: 250000
Revised Variable cost per unit: 160+25 =185 CM per unit: 250-185 = 65 Break even
units: Fixed cost / CM per unit 250000 /65 = 3846 units Req 5: Total cost of
150000 units Mmaterial 15,000,000 Labour 4500000 Variable manufacturing OH
4500000 fixed manufacturing OH 70000 Selling oh 1500000 Fixed selling oh 250000
Total cost of 150000 units 25,820,000 Add: Desired profits 300,000 Target sales
26,120,000 Divide: Number of units 150,000 Selling price per unit 174.13