Math 203 Practice Final Exam

Please work out all of the following problems.  Credit will be given based on the progress that you make towards the final solution.  Show your work.  No calculators allowed for this page.

Key

Problem 1 

Let

       

 Find an orthogonal matrix P and a diagonal matrix D with A  =  PDP-1.

Problem 2

Use the definition of the determinant to find the determinant of  

        Matrix:  [(0 3 0 5), (-9 -8 2 7), (1 -3 0 4), (-4 -1 0 6)

Problem 3

Find the inverse of A if

         




Calculators are permitted on this part

 

Problem 4

Consider the matrix

 

          

A.     Determine the rank of A.

B.   Find a basis for the null space of A.

C.     Find a basis for the column space of A using columns of A.

Problem 5       Let    be defined by

       

A.    Prove that L is a linear transformation.

B.   Let S = {x2 + x, x2 + 1, x} and T = {(1,1), (1,2)} be bases for P2 and R2.  Find the matrix for L with the bases S and T. 

 

Problem 6 

Show that the set

       

is a basis for M2x 2.

 

Problem 7 Use matrices to find the unknown currents in the given circuit.

         

 

Problem 8

Graph the equation and write the equation in standard form.

        4x2 + 2xy + 4y2  =  15

Problem 9

One of the following is a subspace of the space of differentiable functions. 

            I.   {f | f(0) f (0)  =  1}          II. {f | f(1)  =  f (1)} 

 

A.     Determine which is not a subspace and explain why.

B.     Prove that the other one is a subspace. 

 

Problem 10

Prove that if A is a matrix such that A2 = 0 then 0 is an eigenvalue for A.

 

Problem 11

Answer the following true or false. If it is true, explain why.  If it is false explain why or provide a counter example.

A.    If S  =  {v1, v2} is a linearly independent set of vectors in R3 and v3 is not in the span of S, then  {v1, v2, v3} is a basis for R3.

   

B.     Every orthonormal set of five vectors in R5 is a basis for R5.

C.     Let A and B be matrices such that A2v = a, B2v = b, and ABv = c.  Then

(A + B)2 v  =  a + b + 2c
  

      D.  If T is a 1-1 linear transformation from P3 to M2x2, then {T(1), T(1+t), T(1+t2), T(1+t3)} is a basis for M2x2.

      E.   There exists a matrix A that is diagonalizable but not orthogonally diagonalizable.

      F.   If the determinant of a matrix A is greater than 1, then ||Ax|| > ||x|| for all nonzero vector x.