McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 1: Solutions
Note: For ease of reference, the axioms of scalar multiplication of a vector space over
K will be denoted on this assign
McGill University
Math 247B: Linear Algebra
Solution Sheet for Assignment 2
1. (a) 0 W since 0n = 0 for all n = 0n+3 = 0n+2 + 0n = 0 for all n. If x, y W , a, b F we have (ax + by)n+3 =
axn+3 + byn+3 = a(xn+2 + xn ) + b(yn+2 + yn ) = axn+2 + byn+2 + axn +
Inner Product Spaces: Part 1
Let V be a real or complex vector space, i.e., a vector space over R or C. An inner product
on V is a function of V V into R if V is real and into C if V is complex such that, denoting the
value of this function on the pair (u
McGill University
Math 247B: Linear Algebra
Solution Sheet for Assignment 4
1. (a) Let a, b, c be scalars with a sin(x) + b sin(2x) + c sin(3x) = 0 for all x R. Setting x = /2, we get a c = 0.
Dierentiating both sides of the above dependence relation, we
McGill University
Math 247B: Linear Algebra
Solution Sheet for Assignment 3
1. The following generating sets are all linearly independent and hence bases for the subspace they span.
U1 = Span(2, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)
U2 = Span(1, 1, 0, 0),
McGill University
Math 247B: Linear Algebra
Solution Sheet for Assignment 6
1. If X =
x
1
1
,A=
the system is
y
5 3
dX
dt
= AX whose solution is X = eAt X(0). The characteristic matrix of A is
2 + 2 + 2 which has the distinct complex roots 1 + i, 1 i with
McGill University
Math 247B: Linear Algebra
Solution Sheet for Assignment 1
1. (a) R(A B) = R(A) R(B) since y R(A B) (x A B) (x, y) R
(x A) (x, y) R or (x B) (x, y) R y R(A) or y R(B) y R(A) R(B).
(b) R(A B) = R(A) R(B) in general since in the case R = c
Notes on Linear Operators
Theorem 1. Let T : U V be a linear mapping. Then U is nite-dimensional i Ker(T) and
Im(T) are nite-dimensional in which case
dim(U ) = dim(Ker(T) + dim(Im(T).
Proof. () If V is nite-dimensional then so is Ker(T) since a subspace
McGill University
Math 247B: Linear Algebra
Midterm Test
Attempt all questions
1. Let V = RR be the vector space of real valued functions on the real line.
(a) Show that W1 = cfw_f V | f (x) = f (x2 ) for all x R is a subspace of V . Find a linear
operato
The Jordan Canonical Form: Part 2
We now give the proof of the theorem on the Jordan canonical form.
Proof. Without loss of generality, we can assume that the minimal polynomial of T is
( a1 )k1 ( a2 )k2 ( am )km = 0.
By the primary decomposition theorem,
The Jordan Canonical Form: Part 1
Let V be a nite-dimensional vector space over a eld K and let T be a linear operator on V
which satises a polynomial identity of the form
(T a1 )k1 (T a2 )k2 (T am )km = 0
with a1 , a2 , . . . , am distinct scalars. Such
The Decomposition Theorem
The aim of this section is to prove the following theorem
Theorem 1 (Decomposition Theorem). Let V be a vector space over a eld K and let T be a
linear operator on V . If a1 , a2 , ., am are distinct scalars and k1 , k2 , ., km N
Inner Product Spaces: Part 3
Let V be a nite-dimensional inner product space and let T be a linear operator on V . If f is an
orthonormal basis of V , we let T be the linear operator on V such that [T ]f = [T ] . Then, if g is
f
any other orthonormal basi
Inner Product Spaces: Part 2
Let V be an inner product space. We let K = R or C according as V is real or complex. A
sequence of vectors u1 , u2 , u3 , . is said to be orthogonal if < ui , uj >= 0 for i = j. If, in addition,
we have |ui | = 1 for all i th