McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 3: Solutions
1. Let n N and let Pn (K) be the vector space of all polynomials
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 6
You should carefully work out all problems. However, you only have to hand i
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 4
You should carefully work out all problems. However, you only have to hand i
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Midterm Solutions
1. (a) Carefully state all axioms of scalar multiplication for a vector
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 7
This assignment is NOT to be handed in. Nonetheless, you should carefully wo
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 2
You should carefully work out all problems. However, you only have to hand
i
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 9: Solutions
3 2 4
1. Let A = 4 3 8.
2 2 5
(a) Let x0 := (0, 1, 0)t . Compute
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 6: Solutions
1. Let V be a vector space over K, let U be a subspace of V and l
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 3
You should carefully work out all problems. However, you only have to hand
i
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 1
You should carefully work out all problems. However, you only have to hand
i
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 2: Solutions
1. Let V be a vector space over K, let k K and v V . Prove the fo
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 8: Solutions
5 4 11
2 .
1. Let A = 0 1
2 2 5
i 0 0
(a) Show that A is similar
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 7: Solutions
1. Let V be an n-dimensional vector space over K and L : V V be l
Linear Algebra
Done Wrong
Sergei Treil
Department of Mathematics, Brown University
Copyright c Sergei Treil, 2004, 2009, 2011, 2014
Preface
The title of the book sounds a bit mysterious. Why should an
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2018
Instructor: Dr. Axel Hundemer
Office: 1128 Burnside Hall
Email: [email protected]
O
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 5: Solutions
1. Let V and W be vector spaces over K where V is finite dimensio
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 4: Solutions
1. Let V be a finite dimensional vector space over K and let U an
McGill University
Department of Mathematics and Statistics
MATH 247 Honours Applied Linear Algebra
Winter 2017
Assignment 5: Solutions
1. Let V and W be vector spaces over K where V is finite dimensio
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
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
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 )k
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
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 )
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
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
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 +
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,
McGill University
Math 247B: Linear Algebra
Solution Sheet for Assignment 5
1. The vectors
1
f1 = 2 ,
1
1
f2 = 0 ,
1
1
f2 = 1
1
form a basis for V = R31 and
e1 =
1
1
1
f1 + f2 f3 ,
4
4
2
e2 =
1
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.
Dierent