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

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