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1. Let F be a subfield of the complex numbers (or, a field of characteristic zero). Let V be a finite-dimensional vector space over F. Suppose that E1,...,Ek are projections of V and that E1+...+Ek = I. Prove that EiEj = 0 for i not equal to j (Hint: use the trace function and ask yourself what the trace of a projection is).Hmm...the trace of a projection is its rank? But how does that help?
2. Give an example of two 4x4 nilpotent matrices which have the same minimal polynomial (they necessarily have the same characteristic polynomial) but which are not similar.
3. Let V be an n-dimensional vector space, and let T be a linear operator on V. Suppose that T is diagonalizable.
a) If T has a cyclic vector, show that T has n distinct characteristic values.
b) If T has n distinct characteristic values, and if {a1,...,an} is a basis of characteristic vectors for T, show that a = a1+...+an is a cyclic vector for T.
Describe an approach to teaching the base-ten place-value system that incorporates the use of fractions and percentages. For example, consider how you could show your students single popsicle sticks, popsicle sticks bundled in tens, popsicle stick..
A 5000-lb wrecking ball hangs from a 20-ft cable of density 5 lb/ft attached to a crane. Calculate the work done if the crane lifts the ball from ground level to 20 ft in the air by drawing in the cable.
Probability : Counting Principle, A social security number has 9 digits. How many different S.S numbers are possible if:
An oil tank ruptures at time t=0 and oil leaks from the tank at a rate of r(t)=100(e^(-.01t)) liters per minute. How much oil leaks out during the first hour?
Estimate the relative error in the calculated surface area.
An object is thrown upward from the top of a 112-foot building with initial velocity of 96 feet per second.the height of h of the object after t seconds is given by the quadratic equation h =-16t^2+96t+112 when will the object hit the ground?
Suppose Motorola wishes to estimate the mean talk time for its V505 camera phone before the battery must be recharged. In a random sample of 50 phones, the sample mean talk time was 347 minutes.
a garden table and a bench cost combined. The garden table costs more than the bench. What is the cost of the bench?
Computer analysis showed that the surface of a certain drumlin can be approximated by y= 10(1-0.0001x^2) revolved 180 about the x-axis from x= -100 to 100.
A rancher has 400 feet of fencing and wants to build a corral that's divided into three equal rectangles. What length and width will maximize the area?
a rectangular piece of cardboard, whose area is 320 square cm, is made into an open box by cutting a 2-cm square from each corner and turning up the sides. if the box is to have a volume of 384 cubic cm, what size cardboard should you start with?
A representative is to be selected from each of 3 departments in a small college. There are 7 people in the first department, 5 in the second department, and 4 in the third department.
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