Math 2568, Sec 001 Spring 2020
Homework 2 Due Tuesday, February 11, 2020
Instructions:
• Complete each of the problems to the best of your ability. Show all work that leads to your final answer. Any explanations or justifications should be written out in full sentences and (reasonably) correct grammar.
• Only submit your final product. Scratch work should be worked separately and then recopied neatly onto standard letter-sized paper before submission. Your assignment should be stapled with your name clearly labelled on each page. Your work should be legible and the problems should be in the correct order. Do not make me hunt for problems or their supporting work!
• Use the Gauss’s method and back substitution to solve any systems of equations.
Calculator Use:
You may use the basic functions on your calculator to check your arithmetic, but you must show the steps for Gauss’s method in order to receive full credit.
Problems:
The following problems relate to sections One.I.3, One.III.1-2
1. If the homogeneous linear system:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0
has only the trivial solution, what can you say about the solutions of the related nonhomogeneous system given below?
a1x + b1y + c1z = 3
a2x + b2y + c2z = 7
a3x + b3y + c3z = 11
Hint: Is the nonhomogeneous system consistent? How do you know? If the system is consistent, how many solutions will it have?
2. Determine whether the matrix is singular or nonsingular. Show your work. 1 2 32 5 3 −1 −5 5
Math 2568, Sec 001 Spring 2020
3. Consider the set spanned by {~v1, ~v2} =
2−3
1
, 03 −3
. (a) Show that ~0 =
00 0
is in Span {~v1, ~v2}. (b) Identify 5 nonzero vectors in Span {~v1, ~v2}.
(c) Determine whether
1−5 −2
is in Span {~v1, ~v2}. Show your work and/or explain your answers. (d) Determine whether
63 −9
is in Span {~v1, ~v2}. Show your work and/or explain your answers.
4. Solve the linear system using Gauss-Jordan reduction. Identify the reduced echelon form of the augmented matrix and then express the solution set in parametric vector form. Show your work.
t − 2u − v + 3w = 0 −2t + 4u + 5v − 5w = 4 3t − 6u − 6v + 7w = 1
5. (a) List the reduced echelon forms that are possible for 2× 3 matrices. (Hefferon: One.III.1, #14c)
(b) Find two 2× 3 reduced echelon form matrices that have leading entries in the same columns but are not row equivalent. (Hefferon: One.III.2, #20)
6. Explain why any two nonsingular 3×3 matrices are row equivalent. Is it true that any two singular 3×3 matrices are row equivalent? (Hefferon: One.III.2, #21)
7. Determine which of the matrices are row equivalent. Show your work and/or explain your answers.
(a) ( 1 3
2 4
) (b)
( 1 5
2 10
) (c)
( 1 −1 3 0
) (d)
( 2 6
4 10
) (e)
( 0 1
−1 0
)
