Download Homework 1 – Due 1/17 Math 2568

Homework 1 – Due 1/17
Math 2568
Spring 2014
Oguz Kurt
Problem 1
TRUE/FALSE: Any system of linear equations has at least one solution. Explain yourself!
Problem 2
Solve the following system, using Gaussian or Gauss-Jordan elimination using the following steps:
x
2x
−x
−
−
+
4y
8y
4y
−
+
−
z
z
2z
+
−
+
w
4w
5w
=
=
=
3
9
−6
• Write the corresponding augmented matrix [A | !b]!
• Use elementary row operations to find a row echelon form of [A | !b]!
• Use your row echelon form to write the corresponding system of linear equations!
• Use back-substitution to find the set of all solutions!
Problem 3
Solve the following system of linear equations. Give the number
variables in the process.:
3w + 8x − 18y + z =
w + 2x − 4y
=
w + 3x − 7y + z =
of leading terms and number of free
35
11
10
Problem 4
Describe when the augmented matrix
solutions! Here are some helpful hints:
!
k
1
2
2k
"
1
has no solution, unique solution and infinitely many
1
• Use elementary row operations to obtain a row echelon form.
• Avoid using row operations of the form Ri : f (k)Ri or Ri : Ri +
important?
g(k)
h(k) Rj .
Why do you think this is
– Any equation of the form 0 = ! where ! "= 0 implies NO solution.
– If there is a solution, any ZERO row implies infinitely many solutions.
– Otherwise, you have a unique solution.
• Decide which k values give which outcomes!
Problem 5

1
Describe when the augmented matrix 1
2
many solutions!
1
1
4 −1
−1 4

2
k  has no solution, unique solution and infinitely
k2
Problem 6
 
a
For what set of numbers  b  is the following system consistent (that is, it has a solution)?
c
4x1 + x2 + 6x3 = a
2x1 − 3x2 + 3x3 = b
2x1 + 7x2 + 3x3 = c
Hint: You need to get an equation on a, b, c as your answer.