YORK UNIVERSITY
SC/MATH 2022 M
WRITTEN ASSIGNMENT I
Name
Student number
The total number of points is 95.
1. (5+2+2+2+2+2 points) (a) Determine if X ∈ span{Y, Z} where
X = [8
3
− 13
21]T ,
Y = [2
−3
1
5]T ,
Z = [−1
Date: July 03, 2020, Crowdmark submission due date July 13, 2020.
1
0
2
− 3]T .
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YORK UNIVERSITY SC/MATH 2022 M WRITTEN ASSIGNMENT I
(b) In each case below either show that the statement is True or give an example
showing that it is False.
(i) If U 6= Rn is a subspace of Rn and X + Y ∈ U, then X and Y are both in U.
(ii) If U is a subspace of Rn and rX ∈ U for all r ∈ R, then X ∈ U.
(iii) If U is a subspace of Rn and X ∈ U, then −X ∈ U.
(iv) If X ∈ U = span{Y, Z}, then U = span{X, Y, Z}.
(v) If {x1 , x2 , . . . xk , xk+1 , . . . xn } is a basis of Rn and U = span{x1 , . . . xk } and
V = span{xk+1 . . . xn }, then U ∩ V = {0}.
YORK UNIVERSITY
SC/MATH 2022 M
WRITTEN ASSIGNMENT I
3
2. (5+5 points) (a) Find a basis and the dimension of the following subspace U of
R4 : U = {[a a + b a + 2b b]T |a, b ∈ R}.
(b) Determine if the following set of vectors is a basis in R3 .
{[−1
1
− 1]T ,
[1
−1
2]T ,
[0
0
1]T }.
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YORK UNIVERSITY SC/MATH 2022 M WRITTEN ASSIGNMENT I
3. (2+2+2+2+2 points) (a) In each case below show that the statement is True
or give an example showing that it is False.
(i) If {X, Y } is independent in Rn , then {X, Y, X + Y } is independent.
(ii) If {X, Y, Z} is independent in Rn , then {Y, Z} is independent.
(iii) If {Y, Z} is dependent in Rn , then {X, Y, Z} is dependent.
(iv) If A is a 5 × 8 matrix with rank A = 4, then dim(null A) = 3.
(v) If A and B are similar matrices, then so are AT and B T .
YORK UNIVERSITY
SC/MATH 2022 M
WRITTEN ASSIGNMENT I
4. (10 points) Determine the rank of A and find bases for row A and col A.


2 −1 1
 −2
1 1 

A=
 4 −2 3 
−6
3 0
5
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YORK UNIVERSITY SC/MATH 2022 M WRITTEN ASSIGNMENT I
5. (5+5 points) (a) Let A be an n × n matrix. Show that A2 = 0 if and only if
col A ⊆ null A.
(b) Find a matrix A for which col A = null A.
YORK UNIVERSITY
SC/MATH 2022 M
WRITTEN ASSIGNMENT I
6. (10+10 points) (a) Let {X, Y } be a basis of R2 and let
a b
A=
c d
Show that A is invertible if and only if {aX + bY, cX + dY } is a basis in R2 .
7
8
YORK UNIVERSITY SC/MATH 2022 M WRITTEN ASSIGNMENT I
(b) The following is a Markov (migration) matrix for 3 locations 1, 2, and 3.


.6 0 .1
A =  .2 .8 0 
.2 .2 .9
Suppose initially there are 100 residents in location 1, 200 residents in location 2,
and 400 residents in location 3. Find the populations in the three locations after a
long time.
YORK UNIVERSITY
SC/MATH 2022 M
WRITTEN ASSIGNMENT I
9
7. (5+5 points) (a) Use the Gram-Schmidt algorithm to find an orthogonal basis
of the subspace U = span {[1 1 1]T , [0 1 1]T } of R3 .
(b) Show that
B = {[1
2
3]T ,
[−1
−1
1]T ,
[5
−4
1]T }
is an orthogonal basis of R3 and find the Fourier expansion of X = [a
this basis.
b
c]T in
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YORK UNIVERSITY SC/MATH 2022 M WRITTEN ASSIGNMENT I
8. (10 points) Show that the matrix


1 0 1
A =  0 1 0  is diagonalizable but
0 0 2
Find P such that P −1 AP is diagonal.

1
B= 0
0
1
1
0

0
0 
2
is not.