Question

Exercise 2.4 Assume that a system Ax = b of linear equations has at least two distinct solutions y and z.

a. Show that x_{k} = y+k(y−z) is a solution for every
k.

b. Show that x_{k} = x_{m} implies k = m. [Hint:
See Example 2.1.7.]

c. Deduce that Ax = b has infinitely many solutions.

Answer #1

2.4. Let the system Ax = b of linear equations have at least two distinct solutions y and z. Then Ay = b and Az = b.

a. Let k be an arbitrary scalar. Now, if x_{k}
=y+k(y−z),then Ax_{k}= A(y+k(y−z)) = Ay +kA(y-z) =
b+k(Ay-Az) = b+k(b-b) = b. Hence, x_{k} = y+k(y−z) is a
solution to Axx = b for all k.

b. If x_{k} = x_{m}, then y+k(y−z) = y+m(y−z)
or, k(y-z) = m(y-z) so that k = m ( as y ≠ z).

c. As per part (b) above, if the equation Ax = b has two
distinct solutions y and z , then x_{k} = y+k(y−z) is also
a solution for every k. This implies that the equation Ax = b has
infinite solutions.

Find the values of a and b for which the following system of
linear equations is (i) inconsistent; (ii) has a unique solution;
(iii) has infinitely many solutions. For the case where the system
has infinitely many solutions, write the general solution.
x + y + z = a
x + 2y ? z = 0
x + by + 3z = 2

Determine the value of k such that the following system of
linear equations has infinitely many solutions, and then find the
solutions. (Express x, y, and z in terms of the parameters t and
s.) 3x − 2y + 4z = 9 −9x + 6y − 12z = k k = (x, y, z) =

4. [10] Consider the system of linear equations
x + y + z = 4
x + y + 2z = 6
x + y + (b2 − 3)z = b + 2
where b is an unspecified real number. Determine, with
justification, the values of b (if any) for which the system
has
(i) no solutions;
(ii) a unique solution;
(ii) infinitely many solutions.

A: Determine whether the system of linear equations has one and
only one solution, infinitely many solutions, or no solution.
3x - 4y = 9
9x - 12y = 18
B: Find the solution, if one exists. (If there are infinitely
many solutions, express x and y in terms of parameter t. If there
is no solution, enter no solution.)
(x,y)= ?

1)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express your answer in terms
of the parameters t and/or s.)
x1
+
2x2
+
8x3
=
6
x1
+
x2
+
4x3
=
3
(x1,
x2, x3)
=
2)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express...

For a real number "a", consider the system of equations:
x+y+z=2
2x+3y+3z=4
2x+3y+(a^2-1)z=a+2
Which of the following statements is true?
A. If a= 3 then the system is inconsistent.
B. If a= 1 then the system has infinitely many solutions.
C. If a=−1 then the system has at least two distinct
solutions.
D. If a= 2 then the system has a unique solution.
E. If a=−2 then the system is inconsistent.

A linear system of equations Ax=b is known, where A is a matrix
of m by n size, and the column vectors of A are linearly
independent of each other. Please answer the following questions
based on this assumption, please explain it, thank you~.
(1) To give an example, Ax=b is the only solution.
(2) According to the previous question, what kind of inference
can be made to the size of A at this time? (What is the size of...

4. Suppose that we have a linear system given in matrix form as
Ax = b, where A is an m×n matrix, b is an m×1 column vector, and x
is an n×1 column vector. Suppose also that the n × 1 vector u is a
solution to this linear system. Answer parts a. and b. below.
a. Suppose that the n × 1 vector h is a solution to the
homogeneous linear system Ax=0.
Showthenthatthevectory=u+hisasolutiontoAx=b.
b. Now, suppose that...

The augmented matrix represents a system of linear equations in
the variables x and y.
[1 0 5. ]
[0 1 0 ]
(a) How many solutions does the system have: one, none, or
infinitely many?
(b) If there is exactly one solution to the system, then give
the solution. If there is no solution, explain why. If there are an
infinite number of solutions, give two solutions to the system.

q.1.(a)
The following system of linear equations has an infinite number
of solutions
x+y−25 z=3
x−5 y+165 z=0
4 x−14 y+465 z=3
Solve the system and find the solution in the form
x(vector)=ta(vector)+b(vector)→, where t is a free
parameter.
When you write your solution below, however, only write the part
a(vector=⎡⎣⎢ax ay az⎤⎦⎥ as a unit column vector with the
first coordinate positive. Write the results accurate to
the 3rd decimal place.
ax =
ay =
az =

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 15 minutes ago

asked 17 minutes ago

asked 31 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago