Homework 11
Homework
AME 525, Fall 2025
AME
2026
Due:Friday
FridaySeptember
September 11,
12, 2026
2025 at
11:59 p.m.
Due:
at 11:59
p.m.
1.1.2
Solve the system by using elementary row operations on the equations or on the augmented
matrix. Follow the systematic elimination procedure described in section 1.1 of the textbook.
2x1 + 4x2 = →4
5x1 + 7x2 = 11
1.1.12
Solve the system:
x1 → 3x2 + 4x3 = →4
3x1 → 7x2 + 7x3 = →8
→4x1 + 6x2 + 2x3 = 4
1.1.21
Do the three lines
x1 → 4x2 = 1
2x1 → x2 = →3
→x1 → 3x2 = 4
have a common point of intersection? Explain.
1.1.22
Do the three planes
x1 + 2x2 + x3 = 4
x2 → x3 = 1
x1 + 3x2 = 0
have at least one common point of intersection? Explain.
1
1.1.38
Suppose a, b, c, d are constants such that a ↑= 0 and the system below is consistent for all
possible values of f and g. What can you say about the numbers a, b, c, d? Justify your
answer.
ax1 + bx2 = f
cx1 + dx2 = g
1.2.3
Row reduce the matrix to reduced echelon form. Circle the pivot positions in the final matrix
and in the original matrix, and list the pivot columns.
1 2 3 4
4 5 6 7
6 7 8 9
1.2.8
Find the general solutions of the system whose augmented matrix is given below.
%
&
1 4 0 7
2 7 0 11
1.2.24
Choose h and k such that the system has (a) no solution, (b) a unique solution, and (c)
many solutions. Give a separate answer for each part.
x1 + 4x2 = 5
2x1 + hx2 = k
1.3.6
Write a system of equations that is equivalent to the given vector equation.
% &
% &
% & % &
→2
8
1
0
x1
+ x2
+ x3
=
3
5
→6
0
1.3.12
Determine if ωb is a linear combination of ωa1 , ωa2 , ωa3 .
1
0
2
ωa1 = →2 , ωa2 = 5 , ωa3 = 0 ,
2
5
8
2
→5
ωb = 11
→7
1.3.18
Let
1
ωv1 = 0 ,
→4
→5
ωv2 = 1 ,
7
h
ωy = →1 .
→5
For what value(s) of h is ωy in the plane generated by ωv1 and ωv2 ?
1.3.34
Let
2
0 6
A = →1 8 5 ,
1 →2 1
10
ωb = 3 .
3
Let W be the set of all linear combinations of the columns of A.
(a) Is ωb in W ?
(b) Show that the third column of A is in W .
1.4.12
Given A and ωb, write the augmented matrix for the linear system that corresponds to the
matrix equation Aωx = ωb. Then solve the system and write the solution as a vector.
1
2 1
0
ω
A = →3 →1 2 , b = 1
0
5 3
→1
1.4.15
Let
%
&
3 →4
A=
,
→6 8
% &
ωb = b1 .
b2
Show that the equation Aωx = ωb does not have a solution for all possible ωb, and describe the
set of all ωb for which Aωx = ωb does have a solution.
1.4.22
Let
0
ωv1 = 0 ,
→2
0
ωv2 = →3 ,
8
Does {ωv1 , ωv2 , ωv3 } span R3 ? Why or why not?
3
4
ωv3 = →1 .
→5
1.4.46
Let A be a 5 ↓ 3 matrix, let ωy ↔ R3 , and let ωz ↔ R5 . Suppose Aωy = ωz . What fact allows you
to conclude that the system Aωx = 4ωz is consistent?
1.5.5
Follow the method of Examples 1 and 2 (section 1.5 of the textbook) to write the solution
set of the given homogeneous system in parametric vector form.
x1 + 3x2 + x3 = 0
→4x1 → 9x2 + 2x3 = 0
→3x2 → 6x3 = 0
1.5.19
Follow the method of Example 3 (section 1.5 of the textbook) to describe the solutions of
the following system in parametric vector form. Also, give a geometric description of the
solution set and compare it to that in Exercise 5.
x1 + 3x2 + x3 = 1
→4x1 → 9x2 + 2x3 = →1
→3x2 → 6x3 = →3
4
0
