Linear Algebra/Describing the Solution Set/Solutions

SolutionsEdit

This exercise is recommended for all readers.
Problem 1

Find the indicated entry of the matrix, if it is defined.

 
  1.  
  2.  
  3.  
  4.  
Answer
  1.  
  2.  
  3.  
  4. Not defined.
This exercise is recommended for all readers.
Problem 2

Give the size of each matrix.

  1.  
  2.  
  3.  
Answer
  1.  
  2.  
  3.  
This exercise is recommended for all readers.
Problem 3

Do the indicated vector operation, if it is defined.

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
Answer
  1.  
  2.  
  3.  
  4.  
  5. Not defined.
  6.  
This exercise is recommended for all readers.
Problem 4

Solve each system using matrix notation. Express the solution using vectors.

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
Answer
  1. This reduction
     
    leaves   leading and   free. Making   the parameter, we have   so the solution set is
     
  2. This reduction
     
    gives the unique solution  ,  . The solution set is
     
  3. This use of Gauss' method
     
    leaves   and   leading with   free. The solution set is
     
  4. This reduction
     
    shows that the solution set is a singleton set.
     
  5. This reduction is easy
     
    and ends with   and   leading, while   and   are free. Solving for   gives   and substitution shows that   so  , making the solution set
     
  6. The reduction
     
    shows that there is no solution— the solution set is empty.
This exercise is recommended for all readers.
Problem 5

Solve each system using matrix notation. Give each solution set in vector notation.

  1.  
  2.  
  3.  
  4.  
Answer
  1. This reduction
     
    ends with   and   leading while   is free. Solving for   gives  , and then substitution   shows that  . Hence the solution set is
     
  2. This application of Gauss' method
     
    leaves  ,  , and   leading. The solution set is
     
  3. This row reduction
     
    ends with   and   free. The solution set is
     
  4. Gauss' method done in this way
     
    ends with  ,  , and   free. Solving for   shows that   and then substitution   shows that   and so the solution set is
     
This exercise is recommended for all readers.
Problem 6

The vector is in the set. What value of the parameters produces that vector?

  1.  ,  
  2.  ,  
  3.  ,  
Answer

For each problem we get a system of linear equations by looking at the equations of components.

  1.  
  2. The second components show that  , the third components show that  .
  3.  ,  
Problem 7

Decide if the vector is in the set.

  1.  ,  
  2.  ,  
  3.  ,  
  4.  ,  
Answer

For each problem we get a system of linear equations by looking at the equations of components.

  1. Yes; take  .
  2. No; the system with equations   and   has no solution.
  3. Yes; take  .
  4. No. The second components give  . Then the third components give  . But the first components don't check.
Problem 8

Parametrize the solution set of this one-equation system.

 
Answer

This system has   equation. The leading variable is  , the other variables are free.

 
This exercise is recommended for all readers.
Problem 9
  1. Apply Gauss' method to the left-hand side to solve
     
    for  ,  ,  , and  , in terms of the constants  ,  , and  . Note that   will be a free variable.
  2. Use your answer from the prior part to solve this.
     
Answer
  1. Gauss' method here gives
     
    leaving   free. Solve:  , and   so  , and   Therefore the solution set is this.
     
  2. Plug in with  ,  , and  .
     
This exercise is recommended for all readers.
Problem 10

Why is the comma needed in the notation " " for matrix entries?

Answer

Leaving the comma out, say by writing  , is ambiguous because it could mean   or  .

This exercise is recommended for all readers.
Problem 11

Give the   matrix whose  -th entry is

  1.  ;
  2.   to the   power.
Answer
  1.  
  2.  
Problem 12

For any matrix  , the transpose of  , written  , is the matrix whose columns are the rows of  . Find the transpose of each of these.

  1.  
  2.  
  3.  
  4.  
Answer
  1.  
  2.  
  3.  
  4.  
This exercise is recommended for all readers.
Problem 13
  1. Describe all functions   such that   and  .
  2. Describe all functions   such that  .
Answer
  1. Plugging in   and   gives
     
    so the set of functions is  .
  2. Putting in   gives
     
    so the set of functions is  .
Problem 14

Show that any set of five points from the plane   lie on a common conic section, that is, they all satisfy some equation of the form   where some of   are nonzero.

Answer

On plugging in the five pairs   we get a system with the five equations and six unknowns  , ...,  . Because there are more unknowns than equations, if no inconsistency exists among the equations then there are infinitely many solutions (at least one variable will end up free).

But no inconsistency can exist because  , ...,   is a solution (we are only using this zero solution to show that the system is consistent— the prior paragraph shows that there are nonzero solutions).

Problem 15

Make up a four equations/four unknowns system having

  1. a one-parameter solution set;
  2. a two-parameter solution set;
  3. a three-parameter solution set.
Answer
  1. Here is one— the fourth equation is redundant but still OK.
     
  2. Here is one.
     
  3. This is one.
     
? Problem 16
  1. Solve the system of equations.
     
    For what values of   does the system fail to have solutions, and for what values of   are there infinitely many solutions?
  2. Answer the above question for the system.
     

(USSR Olympiad #174)

Answer

This is how the answer was given in the cited source.

  1. Formal solution of the system yields
     
    If   and  , then the system has the single solution
     
    If  , or if  , then the formulas are meaningless; in the first instance we arrive at the system
     
    which is a contradictory system. In the second instance we have
     
    which has an infinite number of solutions (for example, for   arbitrary,  ).
  2. Solution of the system yields
     
    Here, is  , the system has the single solution  ,  . For   and  , we obtain the systems
     
    both of which have an infinite number of solutions.
? Problem 17

In air a gold-surfaced sphere weighs   grams. It is known that it may contain one or more of the metals aluminum, copper, silver, or lead. When weighed successively under standard conditions in water, benzene, alcohol, and glycerine its respective weights are  ,  ,  , and   grams. How much, if any, of the forenamed metals does it contain if the specific gravities of the designated substances are taken to be as follows?

Aluminum 2.7 Alcohol 0.81
Copper 8.9 Benzene 0.90
Gold 19.3 Glycerine 1.26
Lead 11.3 Water 1.00
Silver 10.8

(Duncan & Quelch 1952)

Answer

This is how the answer was given in the cited source.

Let  ,  ,  ,  ,   be the volumes in   of Al, Cu, Pb, Ag, and Au, respectively, contained in the sphere, which we assume to be not hollow. Since the loss of weight in water (specific gravity  ) is   grams, the volume of the sphere is  . Then the data, some of which is superfluous, though consistent, leads to only   independent equations, one relating volumes and the other, weights.

 

Clearly the sphere must contain some aluminum to bring its mean specific gravity below the specific gravities of all the other metals. There is no unique result to this part of the problem, for the amounts of three metals may be chosen arbitrarily, provided that the choices will not result in negative amounts of any metal.

If the ball contains only aluminum and gold, there are   of gold and   of aluminum. Another possibility is   each of Cu, Au, Pb, and Ag and   of Al.

ReferencesEdit

  • The USSR Mathematics Olympiad, number 174.
  • Duncan, Dewey (proposer); Quelch, W. H. (solver) (Sept.-Oct. 1952), Mathematics Magazine 26 (1): 48