+1(978)310-4246 credencewriters@gmail.com
  

DescriptionCSE 3380: Linear Algebra for CSE
University of Texas at Arlington
Spring 2023
Alex Dillhoff
Assignment 3
Vector Spaces
1) Let H be the set of all points inside and on the unit ball in R3 .
(  x
)
H =  y  : x2 + y 2 + z 2 ≤ 1
z
Show that H is not a subspace of R3 .
2) Let W be the set of all vectors of the form


a − 6b
−3a + 2b .
b
Show that W is a subspace of R3 .
For problems 3 and 4, determine if the given set of vectors is a basis for R3 . If the set is
not a basis, does it span R3 ?
3)
     
2
3 
 1
1 , 2 , 3


1
2
3
4)
       
1
2
−5 
 −2
 0  , −3 ,  3  , −3


−4
2
−1
5
5) Find a basis for the null space of the matrix A.


−2 1
0
A =  0 −3 −2
0 −3 5
6) Find a basis for the column space of the matrix in problem 5.
CSE 3380: Assignment 3
Dillhoff
7) Give an explicit description of the null space for the following matrix.


−6 −3 5
−4 −2 2
−8 −4 0
8) Find a basis for the space spanned by the columns of the given matrix..


2
1 −1
 10 5 −5
−8 −4 4
Linear Transformations
9) Parallel lines are preserved in affine geometry. However, angles and distances are
not. An affine transformation is defined as x 7→ Ax + b, where A is an invertible
matrix. Show that an affine transformation is not linear for b 6= 0.
10) Use homogeneous coordinates to complete the following tasks.
(a) Create a transformation matrix T that rotates points by 90◦ clockwise about
the x-axis centered at the point (1, 2, 1).
(b) Show the result of this transform by computing and plotting T (x), where x =
(−2, 3, 1).
11) Use homogeneous coordinates to complete the following tasks.
(a) Find the transformation matrix M that reverses the transformation of problem
2a.
(b) Show that M correctly reverses the transform by computing and plotting
M (T (x)), where x = (−2, 3, 1).
12) Find the transformation matrix T that satisfies the following:
T (x1 , x2 , x3 ) = (2×1 + x3 , −x2 + 4×3 , x1 + 6×2 , 4×1 )
13) In general, for two matrices A and B, AB 6= BA. For the following sets of transformations, show that either AB = BA or AB 6= BA.

cos θ sin θ
k 0
(a) A =
and B =
, where k ∈ R.
− sin θ cos θ
0 k




cos θ sin θ 0
1 0 Tx
(b) A = − sin θ cos θ 0 and B = 0 1 Ty , where Tx , Ty ∈ R.
0
0
1
0 0 1




k 0 0
1 0 Tx
(c) A = 0 k 0, where k ∈ R, and B = 0 1 Ty , where Tx , Ty ∈ R.
0 0 1
0 0 1
You may submit your work as either a scanned PDF OR you may take pictures of
your homework solutions and combine them into a PDF. Do not submit individual images. Rename your submission as LASTNAME_ID_A3.pdf.
2

Purchase answer to see full
attachment

  
error: Content is protected !!