MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 2; Solutions
[1] Let the eld be Q. Let
3
A = 2
1
1 2
1 1
3 0
We will solve both problems [a] and [b] at once by row-reducing the augmented
possible sequence of elementary row operations is the followi
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 2; due Monday, 16 January 2012
Preamble: Consider a system of m linear equations in n unknowns over a eld F :
A11 x1 + A12 x2 + + A1n xn
A21 x1 + A22 x2 + + A2n xn
.
.
.
Am1 x1 + Am2 x2 + + Amn xn
=
=
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 3; due Monday, 23 January 2012
[1] Let A be an m n matrix and let B be an n p matrix. Then C = AB is an m p matrix. Show
that the columns of C are linear combinations of the columns of A. Explicitly,
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 4; due Monday, 30 January 2012
[1] Let V be a vector space over a eld F . Suppose that two vectors u and v in V are linearly dependent.
Prove that one of them is a scalar multiple of the other.
[2] Pr
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 6; due Monday, 13 February 2012
[1] Let T : C3 C3 be the mapping dened by
T (z1 , z2 , z3 ) = (z1 z2 + 2z3 , 2z1 + z2 , z1 2z2 + 2z3 ).
[a] Verify that T is a linear mapping.
[b] What are the conditio
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 5; due Monday, 06 February 2012
Reminder: The mid-term test will be on Monday, February 6 (the same day that this
assignment is due!) from 7:00pm to 8:50pm, in MC 1085. The mid-term test covers
everyt
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 7; due Monday, 27 February 2012
[1] Let V and W be nite-dimensional vector spaces over a eld F , with dim V = n and dim W = m.
Let T : V W be a linear map.
[a] Prove that ker T = cfw_0 if and only if
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 8; due Monday, 05 March 2012
[1] Let F = Q be the eld of rational numbers, and let A be the following 2 2 matrix over Q:
A=
2
1
1
.
3
For each of the following polynomials f over Q, compute the 2 2 ma
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 9; due Monday, 12 March 2012
Preamble: Unless noted otherwise, K will always denote a commutative ring with identity. Let det
denote the determinant function on n n matrices over K , which we proved e
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 10; due Monday, 19 March 2012
Preamble: We learned two more important theoretical applications of determinants: the formula for
the inverse of a matrix using the adjugate, and Cramers rule. We briey r
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 11; due Monday, 26 March 2012
[1] Let a, b, c be elements of a eld F and let A be the following 3 3 matrix over F :
00c
A = 1 0 b .
01a
[a] Prove that the characteristic polynomial of A is x3 ax2 bx c
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 1; due Monday, 09 January 2012
[1] Let Q be the eld of rational numbers. Let p Z be a positive prime integer. Dene
F = cfw_a + b p ; a, b Q.
The set F is a subset of R. Prove that F is a subeld of R.
MATH 146: Advanced Linear Algebra 1
Mid-term test; Solutions
[1] TRUE or FALSE:
[a] A eld F can always be regarded as a 1-dimensional vector space over itself. TRUE:
The eld F is just F 1 , which is a 1-dimensional vector space over F .
[b] An innite-dime
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 10; Solutions
[1] Consider the following 3 3 matrix A over Q:
1
2
1
3
1
4
1
A = 1
2
1
3
1
3
1
4 .
1
5
1
We have already seen that det A = 2160 . We want to use the adjugate formula for the inverse:
1
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 1; Solutions
[1] We need to prove that F satises all of the eld axioms. Since the addition and multiplication operations are inherited from R, so we need only verify that F is closed under addition an
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 6; Solutions
[1] Let T : C3 C3 be the mapping dened by
T (z1 , z2 , z3 ) = (z1 z2 + 2z3 , 2z1 + z2 , z1 2z2 + 2z3 ).
[a] Let z = (z1 , z2 , z3 ) and w = (w1 , w2 , w3 ) be in C3 . We can compute direc
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 4; Solutions
[1] Let V be a vector space over a eld F . Suppose that two vectors u and v in V are linearly dependent.
t
Then there exist scalars s and t, not both zero such that su + tv = 0. If s = 0,
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 3; Solutions
[1] Let A be an m n matrix and let B be an n p matrix. Then C = AB is an m p matrix. If
1 , . . . , n are the n columns of A, and 1 , . . . , p are the p columns of C , then we want to sh
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 5; Solutions
[1] Let V = F nn be the vector space of all n n matrices over F . We consider the following four
subsets of V :
S = cfw_A V ; Aij = Aji for all i, j = 1, . . . n
K = cfw_A V ; Aij = Aji f
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 8; Solutions
[1] Let F = Q be the eld of rational numbers, and let A be the following 2 2 matrix over Q:
A=
2
1
1
.
3
We can compute easily that
A2 =
2
1
1
3
2
1
1
3
=
3
5
5
,
8
A3 = A2 A =
3
5
5
8
2
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 7; Solutions
[1] [a] Suppose that ker T = cfw_0, and that cfw_v1 , . . . , vk is a linearly independent subset of V . We need
to show that cfw_T (v1 ), . . . , T (vk ) is a linearly independent subse
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 11; Solutions
[1] Let a, b, c be elements of a eld F and let A be the following 3 3 matrix over F :
00c
A = 1 0 b .
01a
[a] The characteristic polynomial is p = det(xI A), which we can compute directl
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 9; Solutions
[1] Consider the following 3 3 matrix A over a eld
1
A = 1
1
F.
p
q
r
p2
q2 ,
r2
where p, q, r K .
[a] We perform Laplace expansion along the rst column. This yields
det A = (1) det
q
r
q
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 12; Solutions
[1] Let V be a nite-dimensional vector space, and let W1 be a subspace of V .
[a] Let n = dim V and k = dim W1 . Choose a basis cfw_v1 , . . . , vk and extend it to a basis cfw_v1 , . .
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 12; due Monday, 02 April 2012
Preamble: Recall that we say a vector space V has a direct sum decomposition V = W1 W2 Wk
if each Wi is a subspace of V , and
Wi (W1 + + Wi1 + Wi+1 + + Wk ) = cfw_0
for a
MATH 146: Linear Algebra 1 (Advanced Level)
Supplementary notes on Lagrange Interpolation
Let F be a eld, and let Pn be the vector space over F of all polynomials of degree at most n with
coecients in F . That is,
Pn = cfw_a0 1 + a1 x + + an xn ; ai F, i
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 11; due Wednesday, 03 April 2013
[1] Let A be a 2 2 matrix over the eld R. Suppose that A is symmetric: AT = A.
[a] Compute the characteristic polynomial of A in terms of the entries of A.
[b] Prove t
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 12; due Monday, 08 April 2013
[1] Let V be an n-dimensional vector space, and let T : V V be a linear operator. Suppose W is
a T -invariant subspace of V . If T is diagonaliable, prove that the restri
MATH 146: Advanced Linear Algebra 1
Mid-term test; Monday, 11 February 2013; 7:00pm 8:50pm
Instructions:
This test consists of 7 questions over 2 pages, worth a total of 55 points.
Answer all questions in the test booklet. The question sheet will not be l
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 6; due Wednesday, 27 February 2013
[1] Let T : R3 R3 be the mapping dened by
T (x1 , x2 , x3 ) = (x1 + 2x2 + x3 , 2x1 x2 + x3 , x1 + 2x2 + x3 ).
[a] Let x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ) be
MATH 146
Assignment 2
Solutions (revised)
Definition. Given a vector space V and two nonempty subsets S1 , S2 V , define the set
addition of S1 and S2 by
def
S1 + S2 = cfw_x1 + x2 : x1 S1 and x2 S2 .
1. In R2 , let S1 denote the x-axis and let S2 denote t
MATH 146
Assignment 3
Solutions
1. Suppose x1 , . . . , xn are distinct vectors in a vector space V and cfw_x1 , . . . , xn is linearly
independent. For i = 1, . . . , n define yi = x1 + x2 + + xi . Prove that cfw_y1 , . . . , yn is
linearly independent
MATH 146
Assignment 1
Solutions
1. Consider the set R+ = cfw_x R : x > 0 equipped with the following addition and
scalar multiplication by Q operations, denoted x y and a x, defined by
def
x y = xy (ordinary multiplication of real numbers)
def
a x = xa