Practice Exam 1B on Matrices and Matrix Calculations | M 340L, Exams of Mathematics

Material Type: Exam; Professor: Schurle; Class: MATRICES/MATRIX CALCULATNS-C S; Subject: Mathematics; University: University of Texas - Austin; Term: Fall 2009;

Typology: Exams

Pre 2010

Uploaded on 05/23/2010

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M340L EXAM 1B 2:00
FALL, 2009
Dr. Schurle
Show all your work on these pages. Be organized and neat. Your work should be your
own; there should be no talking, reading notes, checking laptops, using cellphones, .. . .
1. (16 points) After many row operations you have changed the augmented matrix of a
system of linear equations to the following, which is not yet in reduced echelon form.
Describe all solutions of the system in parametric vector form. DO NOT USE A
CALCULATOR!! SHOW ALL YOUR WORK STEP BY STEP!!!
1 4 0 0 0 2
2 8 4 1 20 34
1 4 3 0 12 20
0 0 2 1 46
pf3
pf4
pf5

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M340L EXAM 1B 2:

FALL, 2009

Dr. Schurle

Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones,....

  1. (16 points) After many row operations you have changed the augmented matrix of a system of linear equations to the following, which is not yet in reduced echelon form. Describe all solutions of the system in parametric vector form. DO NOT USE A CALCULATOR!! SHOW ALL YOUR WORK STEP BY STEP!!!   

  

YOUR SCORE: /

  1. (14 points) Give three different properties of a list of vectors, each equivalent to the list of vectors being linearly dependent. Then choose two of your properties and explain why they are equivalent, that is, explain why a list of vectors that has each one of the properties must also have the other.
  2. (14 points) Suppose A and C are p × p matrices such that AD = Ip. Explain why the equation Ax = b has a solution for every b in Rp^ and then explain what this says about the columns of A.
  1. (14 points) Find the general flow pattern of the network shown in the figure. Assuming that the flows are all nonnegative, what is the largest possible value for x 3?
  1. (14 points) Suppose the linear transformation T from R^2 to R^3 maps u =

[ 5 3

] into  

  and v =

[ 8 5

] into

 

 

(a) Find T (3u + 5v)

(b) Show that Span{u, v} = R^2.

(c) Use parametric vector form to describe the range of T.