Linear Algebra - Homework 1
Due: Saturday July 13, by 11:59pm,
via Gradescope
•Note that you must assign a page to each problem you submit.Gradescope has great YouTube
videos available on how to submit homework.Failure to submit homework correctly will result
in a zero on homework.
•Late homework is not accepted for any reason. Don’t ask.
1. Solve the following systems of equations by constructing the augmented matrix associated to the system
of equations and using elementary row operations to obtain an augmented matrix in REF. Pivots not
equal to one are OK. Clearly list your ERO’s in the order that you are using them. Place a box around
your final answer.
(a)
4x−9y−12z= 1
3x−4y−11z= 2
−x+ 9y−2z= 3
(b)
−7x+ 7y−z= 4
x−11y+ 6z= 5
−2x−7y+ 5z= 6
2. Prove that it is impossible for the equationA⃗x=
⃗
bto have exactly two distinct solutions.
3. Solve the following matrix equations by setting up the augmented matrix and using elementary row
operations to obtain an augmented matrix in REF. Pivots not equal to one are OK. Clearly list your
ERO’s in the order that you are using them. Place a box around your final answer. Note that your answer,
if it exists, should be a vector. If there is no solution, simply state that the equation is “Inconsistent”
(a)
10−2
1−1−1
29−12
⃗x=
7
8
9
(b)
−445
−323
2−7−6
⃗x=
10
11
12
(c)
15 6
32 6
−1 9 7
⃗x=
13
14
15
(d)
8−117
6−10−5
7−11−2
⃗x=
16
17
18
(e)
2−4−9
012
−265
⃗x=
19
20
21
(f)
1−10−4
0−21−1
3−9−11
⃗x=
22
23
24
4. Let
A=
1 1 1
2 4 4
3 7k
.
(a) Find all values ofkfor which the equationA⃗x=
1
2
3
has exactly one solution. (You do not need
to solve the system).
(b) Find all values ofkfor which the equationA⃗x=
1
2
3
has infinitely many solutions. (You do not
need to solve the system).
5. True or false: If true give a reason, if false give a counterexample.
(a) If⃗u=
1
1
1
is perpendicular to⃗vand⃗w, then⃗vis parallel to⃗w.
(b) If⃗uis perpendicular to⃗vand⃗w, then⃗uis perpendicular to⃗v+ 2⃗w.
(c) If⃗uand⃗vare perpendicular unit vectors, then∥⃗u−⃗v∥=
√
2.
6. Consider the following vectors inR
4
.
⃗v
1
=
1
2
−2
1
, ⃗v
2
=
4
0
4
0
, ⃗v
3
=
1
−1
−1
−1
, ⃗v
4
=
1
1
1
1
.
(a) Find all pairs of vectors that are orthogonal.
(b) Find the angle between the pairs of vectors that are not orthogonal. (Use a calculator and leave
your final answer in radians correct to 4 decimal places)
Remark.You may assume that the angle between two vectors of any dimension is obtained by
using the formula
⃗v·⃗w=∥⃗v∥∥⃗w∥cosθ
7. Write each of the following a matrix equation,A⃗x=
⃗
b. Specify what the matrixA, the vector⃗x, and
the vector
⃗
bare.
(a)
x
1
3
−2
8
+x
2
5
0
9
=
2
−3
8
(b)
x
1
−2x
2
+x
3
= 0
2x
2
−8x
3
= 8
−4x
1
+ 5x
2
+ 9x
3
=−9
8. Consider the matrices
A=
−13
24
5−3
, B=
20−1
4−52
, C=
12
−2 1
.
Compute, if possible,AB,BA,AC,CA,BC,CB. If the product does not exists, explain why.
