Math 203 Practice Midterm 3

Please work out each of the given problems.  Credit will be based on the steps towards the final answer.  Show your work.  Do your work on your own paper.

Problem 1

A physicist has plotted the position of a projectile over time.  Based on Newton’s laws, the projectile should ideally travel in a parabolic path.  Use matrices to find the most likely equation of this parabola.  (You may use a calculator, but show the matrices that are being manipulated).

 Time 1 2 3 4 Distance 2 30 20 2

Problem 2

A new species of fish is introduced into the Truckee River.  Initially 2 fish were stocked.  It takes one year for this species to spawn, when each fish averages 3 successful children each year.  (So there are 2 at the beginning, 2 at the end of the first year, 8 at the end of the second year, 14 at the end of the third year, etc.)

A.     Assuming no fish die, set up a recursion relationship that gives the number of fish wn at the end of year n.

Solution

B.     Find a matrix A such that wn-1  =  An-1(w0, w1)T .

Solution

C.     Find a diagonal matrix D such that A is similar to D.

Problem 3

Let V be the subspace of differentiable functions spanned by {ex, e2x, e3x} and let

L:  V --->  V

be the linear transformation with

L(f(x))  =  f ''(x) - 3f '(x) + 2f(x)

A.     Write down the matrix AL with respect to the given basis.

Solution

B.     Find the a basis for the kernel and range of L.

Problem 4

Let W = Span{(1,1,0,1), (0,1,2,3)}.  Find a basis for the orthogonal complement of W.

Problem 5

Let

and T be the affine transformation T(x)  =  Ax + b .  Sketch the image under T of the figure below.

Problem 6

Let  L:  V ---> V  be a linear transformation.  Use the fact that

dim(Ker L) + dim(Range L)  =  dim(V)

to show that if L is one to one then L is onto.

Problem 7

Let A and B be matrices and let v be an eigenvector of both A and B.  Prove that v is an eigenvector of the product AB.

Problem 8

A.    Let A be a 3x3 matrix such that the columns of A form an orthonormal set of vectors.  Then

B.     Let V be the vector space of continuous functions then the expression

<f, g>  =  f(1) + g(1)

defines an inner product on V.