It has been observed that in an egg-production program the ...



342 Extra Credit problems

5. 4 pts into HW

Refer to the application problem on Electric networks on page 22.

(a) Solve problem #20(c) on page 33. Set up the matrix equation of this solution, i.e., write up the matrix version of Ohm’s law.

(b) Prove that this is a linear model. (this is basically true because of Ohm’s law and Kirchoff’s law are linear). Make sure you verbally state those laws in terms of your proof.

6. 5 pts into HW

Find a least squares line for the following data points (1,1), (2,1) and (3,3) two different ways:

(a) using the linear algebra approach of section 5.3

(b) using multivariable calculus. See the main ideas in our book on the bottom of page 255.

7. 4 pts into HW

It has been observed that in an egg-production program the number of low-cholesterol eggs laid per day by the flock seems to be related to the amounts of the two special feed mixes x1 and x2 used. The relationship is expressed by a linear equation of the form y = a x1 + b x2 . Research results are given below:

x1 x2 y

1 0 4

0 1 5

1 1 6

2 1 5

1 3 4

Using least square fit, find the best approximation to a and b.

8. 4 pts into HW

The inner product is a very powerful tool that enables us to control the geometry of Rn. Different inner products on Rn lead to different measures of vector norm, angle and distance – that is, to different geometries. The scalar product leads to Euclidean geometry, some other inner products lead to some non-Euclidean geometries. Consider the following inner product in R2:

= x1y1+ 4 x2y2.

a) Find the norm of the vector (0,1)T.

b) Show that the vectors (1,1)T and (-4,1)T are orthogonal. What would be the angle between these vectors in the Euclidean space?

c) Find the distance between the vectors (1,0)T and (0,1)T. What would be the distance between these vectors in the Euclidean space?

(d) Find the equation of the circle with center at the origin and of radius one in this space. Sketch the circle.

You just developed in this example a geometry that is correct mathematics even though it does not describe the everyday world in which we live. A lot of times mathematical results are not necessarily useful or related to real world concepts or oriented towards applications. BUT, you never know when such pure mathematics will turn out to be useful. This happened to non-Euclidean geometry. Albert Einstein found that this was the type of geometry he needed for describing space-time. He used R4 to represent three spatial dimensions and one time-dimension. But Euclidean geometry is not appropriate when space and time are “mixed”.

9. 7 pts into HW

(read the above first!!!) Consider the following scenario: a pair of twins are separated immediately after birth and one would remain on the Earth and the other would go on a trip to the nearest star to Earth other than the Sun (it is called Alpha Centauri and is about 4 light years away). He would travel with 0.8 the speed of light, and he would return to Earth immediately upon arrival. By the predictions of Einstein’s theory the traveling twin would be younger at the time the twins meet again on Earth. In general, the age difference varies according to how far and how fast the traveler goes.

Special relativity theory involves four coordinates, x1, x2, x3 are the space coordinates and x4 is the time coordinate. None of the inner products on R4 would lead to a geometry that conforms with experimental results, so special relativity theory uses a so called pseudo inner product. (The first requirement for inner products, that is only zero for the zero vector will be violated.) This pseudo inner product used in Minkowski geometry:

= - x1y1- x2y2 - x3y3 + x4y4.

The norm is defined as

║ x ║ = ( |< x, x>| )1/2. (Notice the absolute value!)

and the distance is defined as

d(x,y) = ( | - (x1 – y1 )2 - (x2 – y2 )2- (x3 – y3 )2+ (x4 – y4 ) | )1/2

Let’s use this geometry to describe the behavior of time for the voyage to Alpha Centauri. Draw a space-time diagram such that (for convenience) we assume that Alpha Centauri lies in the direction of the x1 axis from Earth and remember that on the x4 axis are the time coordinates. Thus the twin who remains on the Earth moves only in the positive direction of the x4 axis. The twin on board of the spaceship advances in time and also moves in the direction of increasing x1 and when coming back to Earth he is still advances in time but moves in the direction of decreasing x1.

(a) Find how many years old the twin who remains on the Earth will be when his twin brother arrives back to Earth. (i.e., find the distance between the initial point and the end point along the x4 axis). Remember that Alpha Centauri is 4 light years away from the Earth.

(b) Find the distance traveled by the other twin.

By a postulate of special relativity theory the distance between two points on the path of an observer, (such as the traveling twin), in Minkowski geometry, corresponds to the time recorded by that observer in traveling between the two points.

Using your answer from part (b) and the postulate, find how many years old the traveling twin be when he arrives back to Earth.

(c) Find a counter example to show that the first requirement for inner products, (that is only zero for the zero vector) will be violated .

(d) Show that the other three requirements for inner products will be true in the defined pseudo inner product.

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