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
possib
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 +
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 ar
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 tha
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] Verif
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
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 l
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
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 determin
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
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
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
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
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 .
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
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
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
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 =
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
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
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 po
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
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
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
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
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 po
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 subsp
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 question
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 ).