MATH 203 PRACTICE EXAM I  

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work. 

Key

Problem 1

Let  L:  R² → R³   be a linear transformation such that 

        L (1,4) = (1,-1,3)   and     L (0,2) =  (2,1,4)

Find L(1,0)

Problem 2

Suppose that you want to change the graphic file from the one on the left to the one on the right.  What is the matrix needed to adjust the pixels appropriately?  Assume the center of the picture is the origin.

Problem 3 For each of the two subsets below, determine if it is a subspace of Rⁿ.  If it is a subspace prove that it is.  If it is not a subspace prove that it is not.

1. {(x,y,z) in R³ | x - yz = 0}
2. {(x,y,z,w) in R⁴ | x + 2y = z - w}

Problem 4 

Answer the following true or false and explain your reasoning.

A.     If A and B are n x n matrices and AB  =  0, then either A = 0 or B = 0.

B.     If A is a matrix with A²  =  I_n then either det(A)  =  1 or det(A)  =  -1.  

C.  If the columns of an mxn matrix A form a basis for R^m, then m = n.

D.  If the nullity of an 8x11 matrix A is 6 then the column space of A is a 5 dimensional subspace of R⁸.

E.  If the second column of a 4x4 matrix A does not contain a pivot, then A is not invertible.

F.  If (1,3,4,1)^T and (2,5,0,6)^T are both solutions of Ax = b, then the equation Ax = 0 has a nontrivial solution.

Problem 5 

Let

A.     Find the rank and the nullity of A. 

B.     Find a basis for the Null Space of A.

C.     Find a basis for the Column Space of A.

Problem 6 

  Let  S  =  {v₁, v₂, ..., v_n} be a set of linearly independent vectors and let v be a vector in the span of S.  Prove that v can uniquely be written as a linear combination of elements of S.  That is that prove that if

      v  =  a₁v₁ +a₂v₂ + ... + a_nv_n        and        v  =  b₁v₁ +b₂v₂ + ... + b_nv_n     

then

        a₁  =  b₂, a₂  =  b₂, ... , a_n  =  b_n

Problem 7 Use the definition of span to prove that the vectors shown below span R².

v = (3,5) and w = (2,3)

Problem 8 

Prove that if v and w are solutions to the matrix equation Ax = b and if r + s = 0, then

rv + sw is a solution to the homogeneous equation Ax = 0.

Problem 9

As a consultant for the South Tahoe ski marketing group, your task is to make predictions on the ski pass that locals hold.  You found that this year 55% of locals hold a pass for Heavenly, 30% hold a pass for Sierra-at-Tahoe, and 15% hold a pass for Kirkwood.  Each year, 20% of the Heavenly pass holders defect to Sierra-at-Tahoe and 5% defect to Kirkwood.  10% of the Sierra-at-Tahoe pass holders defect to Heavenly and 20% defect to Kirkwood.  5% of the Kirkwood pass holders defect to Heavenly and 25% defect to Sierra-at-Tahoe.  Assume that every pass holder remains a pass holder the following year and that everyone holds exactly one pass.  Determine what the distribution of pass holders will be in five years.

Problem 10

Use the definition to find the determinant of the following matrix.  Do not use your calculator.

Extra Credit:  Write down one thing that your instructor can do to make the class better and one thing that you want to remain the same in the class.

(Any constructive remark will be worth full credit.)