Vector Spaces

How to prove or counter with example the following statements: (1) If two subspaces are orthogonal, then they are independent. (2) If two subspaces are independet, then they are orthogonal. I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1 and W2 are said to be orthogonal subspaces if for every w1 element W1 and w2 elemet W2 the inner product satisfies (w1, w2)=0 I would appreciate if you could provide proving expalnation on two of above statements.

Shortest Path Problem

1 a. Three cities are at the vertices of and equilateral triangle of unit length. Flying Executive Airlines needs to supply connecting services between these three cities. What is the minimun length of the two routes needed to supply the connecting service? 1 b. Now suppose Flying Executive Airlines adds a hub at the "center" of the equilateral triangle. Show that the length of the routes needed to connect the three cities has decreased by 13%. (Note: It has been shown that no matter how many "hubs" you add and no matter how many points must be connected, you can never save more than 13% of the total distance needed to "span" all the origional points by adding hubs.)

Formulate a Linear Programming model

3. The Heavy Metal Shop has four heavy presses it uses to stamp out prefabricated metal covers and housings for aviation consumer products. All four presses operate differently and are of different sizes. Currently the shop has a contract to produce three aviation products. Due to security reasons, the products are called Alpha, Beta, and Gamma. The contract calls for 400 units of Alpha, 570 units of Beta, and 320 units of Gamma. The time (in minutes) required for each aviation product to be constructed on each press machine is as follows: Machine Product 1 2 3 4 Alpha 35 41 34 39 Beta 40 36 32 43 Gamma 38 37 33 40 Machine 1 is available for 150 hours, machine 2 for 240 hours, mac... click for more

Formulate a Linear Programming model

5. Bob Brown's 40th birthday party promised to be the social event of the year in Illinois. To prepare, Bob stocked up on the following liquors. Liquor On Hand (ounces) Bourbon 52 Brandy 38 Vodka 64 Dry Vermouth 24 Sweet Vermouth 36 Bob decided to mix four drinks for the party: Chaunceys, Sweet Italians, Bourbon on the Rocks, and Russian Martinis. A Chauncey consists of ¼ bourbon, ¼ vodka, ¼ brandy, and ¼ sweet vermouth. A Sweet Italian contains ¼ brandy, ½ sweet vermouth, and ¼ dry vermouth. Bourbon on the Rocks contains only bourbon. Finally, a Russian Martini consists of ⅓ dry vermouth and ⅔ vodka. Each drink contains 4 ounces. Bob's objective is to mix these... click for more

The Employee Credit Union at Directional State University is planning the allocation of funds for the coming year.

The Employee Credit Union at Directional State University is planning the allocation of funds for the coming year. ECU makes four types of loans and has three additional investment instruments. Each loan/investment has a corresponding risk and liquidity factor (on a scale of 0-100, with 100 being the most risky/liquid). The various revenue-producing instruments are summarized in the table below: Instrument Annual Rate of Return (%)Risk Factor Liquidity Factor Automobile loans 8 50 0 Furniture loans 10 60 0 Other secured loans 11 70 0 Unsecured lo... click for more