Scribd
Upload
3 views2 pages

hw03

This document outlines Homework 3 for Math 21b: Linear Algebra, due on February 5 and 9, 2016. It includes problems related to determining the number of solutions for various systems of equations, probabilities of matrix ranks, and constructing systems with specific solution characteristics. Additionally, it discusses the properties of consistent and inconsistent systems and provides equations related to humidity in a herb garden.

Uploaded by

moien
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
3 views2 pages

hw03

This document outlines Homework 3 for Math 21b: Linear Algebra, due on February 5 and 9, 2016. It includes problems related to determining the number of solutions for various systems of equations, probabilities of matrix ranks, and constructing systems with specific solution characteristics. Additionally, it discusses the properties of consistent and inconsistent systems and provides equations related to humidity in a herb garden.

Uploaded by

moien
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

Math 21b: Linear Algebra Spring 2016

Homework 3: The number of solutions


This homework is due on Friday, February 5, respectively on Tuesday February 9, 2016.
 
2 3 4 5 6 

 
3 4 5 6 7 



1 Given A = 4 5 6 7 8 . For each of the vectors ~b given below,
 



5 6 7 8 9 
 


 
6 7 8 9 10
determine

whether

the

system

A~x= ~b has 0, 1 or

∞ many solutions.
 
 1   1   5   0   0 
         
 1   2   6   0   0 
         
         
~ 


 ~ 


 ~ 


 ~ 


 ~
a) b =  1  b) b =  3  c) b =  7  d) b =  0  e) b =  0 



         
 1   2   8   0   0 
         
         
         
1 1 9 0 1

2 Consider the set X of all 2 × 2 matrices with matrix entries 0 or


1. The probability of a set of matrices Y with some property is the
number of matrices in Y divided by the number of matrices in X.
a) What is the probability that the rank of the matrix is 1?
b) What is the probability that the rank of the matrix is 0?
c) What is the probability that the rank of the matrix is 2?

3 As in the previous problem, now also the 2-vector b take randomly


the values 0,1, we can look at all the possible equations Ax = b, where
A, b are obtained with 0 or 1 entries. The probability space has now
64 elements.
a) What is the probability that the system has a unique solution?
b) What is the probability that the system has no solution?
c) What is the probability that the system has infinitely many solu-
tions?
4 Build your own system of equations for three variables or state that
there is none. Your system has to have the form a11 x+a12 y+a13 z = b1 ,
a21 x + a22 y + a23 z = b2 , a31 x + a32 y + a33 z = b3 with all aij nonzero.
a) An example with exactly one solution.
b) An example with no solutions.
c) An example where the solution is a plane.
d) An example where the solution is a line.
e) An example where the solution space is three dimensional.
5 In a herb garden, the soil has the property that at any given point the
humidity is the sum of the neighboring humidities. Samples are taken
on a hexagonal grid on 14 spots. The humidity at the four locations
x = y+z+w+2
y= x+w-3
x, y, z, w is unknown. Solve the equations using
z= x+w-1
w = x+y+z-2
row reduction.
1 -1 -1

2 x y -1

1 z w 0

-4 0 -2

Main properties

A system which has a solution is called consistent. Otherwise


it is called inconsistent.

We have a unique solution to A~x = ~b if and only if rref(A)


has a leading 1 in every column and the system is consistent.
We have no solution if and only if rref(A|b) has a leading 1
in the last column. In all other cases we have infinitely many
solutions.

You might also like