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Spring 2018
Instructor: Dr. Alfred Chong
Assignment 5
READ THE FOLLOWING BEFORE YOU START
This assignment contains 5 pages (including this cover page) and 7 problems. Check to see
if any pages are missing. Enter all requested information on the top of this page, and put
your initials on the top of every page, in case the pages become separated.
You are required to show your work on each problem. The following rules apply:
? Organize your work, in a reasonably neat and coherent way, in the space provided.
Work scattered all over the page without a clear ordering will receive very little credit.
? Mysterious or unsupported answers will not receive full credit. A correct answer,
unsupported by calculations, explanation, or algebraic work will receive no credit;
an incorrect answer supported by substantially correct calculations and explanations
will receive partial credit.
? You can use R to verify your the answers to computational problems. You may want
to use RStudio IDE for R.
Do not write anything on this page(except your name).Math 410 Assignment 5 - Page 2 of 5
Problem Points Score
1 15
2 20
3 15
4 15
5 12
6 17
7 6
Total: 100Math 410 Assignment 5 - Page 3 of 5
1. (15 points) Let A be a matrix of size m × n. Show that the column space C(A) and the
left nullspace N(AT
) are orthogonal in R
m. That is, prove that
C(A) ⊥ N(A
T
).
2. Consider the following matrix
A =
?
?
1 ?2 2 ?2
?3 6 ?4 0
1 ?2 3 ?5
?
? .
(a) (2 points) Find the column space C(A).
(b) (2 points) Find a basis of the column space C(A).
(c) (2 points) Find the nullspace N(A).
(d) (2 points) Find a basis of the nullspace N(A).
(e) (2 points) Find the row space C(AT
).
(f) (2 points) Find a basis of the row space C(AT
).
(g) (2 points) Find the left nullspace N(AT
).
(h) (2 points) Find a basis of the left nullspace N(AT
).
(i) (2 points) Verify the rank-nullity theorem for the column space C(A), the nullspace
N(A), the row space C(AT
), and the left nullspace N(AT
).
(j) (2 points) Verify the orthogonality for the column space C(A), the nullspace N(A),
the row space C(AT
), and the left nullspace N(AT
).
3. (15 points) Let A be a matrix of size m × n with linearly independent rows. Show that
the orthogonal projection ?→x r of ?→x ∈ R
n onto the row space C(AT
) is given by
?→x r = A
T
(AAT
)
?1A
?→x ,
with the projection matrix given by Pr = AT
(AAT
)
?1A so that ?→x r = Pr
?→x , and hence
the orthogonal projection ?→x n of ?→x ∈ R
n onto the nullspace N(A) is given by
?→x n = (I ? A
T
(AAT
)
?1A)
?→x ,
with the projection matrix given by P?
n = I ?Pr = I ?AT
(AAT
)
?1A so that ?→x n = P?
n
?→x .
4. Let
?→b ∈ R
4
. Consider the following matrix
A =
?
?
?
?
1 2 1
0 1 0
1 1 1
?1 ?2 ?3
?
?
?
?
.
(a) (3 points) Determine the condition(s) among b1, b2, b3, b4 ∈ R such that the system
of linear equations A
?→x =
?→b is consistent.Math 410 Assignment 5 - Page 5 of 5
(a) (15 points) Fit a regression plane using the method of least squares.
(b) (2 points) Suppose that a new policyholder of age 45, who has 10 years of driving
experience, joins the auto coverage plan. Find the expected annual number of car
accidents for this policyholder.
7. Which of the following statements are true/false? Justification for each part is required.
(a) (2 points) The dimension of the column space C(A) is the same as the dimension
of the row space C(AT
) which is also the same as the rank of the matrix A.
(b) (2 points) If det(A) = 0, then the left nullspace N(AT
) is the zero subspace {
?→0 }.
(c) (2 points) The sum of the dimensions of the row space C(AT
) and the null space
N(A) equals the number of rows in A.