McGill University
Department of Mathematics and Statistics
MATH 270 Applied Linear Algebra
Fall 2011
Assignment 6: Solutions
1. Consider the linear recurrence relation an+2 = 3an+1 2an .
(a) Find the general solution of this recurrence relation.
(b) Find
Middle of term quiz version 1., Math 270. Thursday Oct 24th. Do all 1
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work 111 Space pmwded. N0 calcuiafcors Show all you? work S 3 KM
, x . . 1 ~1 * w;
1. (10 marks) Prove that the set of matmceg that commute Wlth ( O 3 > a}?
' . . . / iii
is
Middle of term quiz version 4, Math 270. Thursday Oct. 25th. Do all
work in space provided. No calculators. Show all your work
1. (5 marks) Prove or disprove: If V is an inner product space and v V
is xed, then cfw_v < w, v >: w V is a subspace of V .
1
Math 270 nal exam version 1. Thursday Dec. 6th, 2012. 9am - noon.
Do all work in space provided. No calculators. Show all your work, except
for the multiple choice portion.
Multiple choice portion. 3 marks each for correct answers. Please circle
the corre
Middle of term quiz version 1, Math 270. Thursday Oct. 25th. Do all
work in space provided. No calculators. Show all your work
1. Consider the linear transformation F : R2 R2 for which F (1, 1) =
(3, 1) and F (3, 2) = (11, 13)
(a) (5 marks) Find the matri
Namezmderline family name)
Student Number: Signature:
MCGILL U NIV ERSIT Y
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATH 270
Applied Linear Algebra
Examiner: Professor D. Wise Bate: Monday December 5, 2085,
Associate Examiner: Professor J. Loveys Time
McGILL UNIVERSITY
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATH 270
Applied Linear Algebra
Examiner: Professor J. Love 5 Date: Wednesday April 26, 2006
Associate Examine Prof 301' D. Wise Time: 2:00 PM - 5:00PM
/mm¢ W
INSTRUCTIONS
1. Please answer all
Midterm Exam of Applied Linear Algebra - Math 270
March 9, 2011
Notation: Let Pn be the space of real polynomials of degree less or equal than n, e.g.:
P1 = cfw_ax + b, for a, b R, P2 = cfw_ax2 + bx + c, for a, b, c R.
Problem 1
(2 pts.)
Find the (complex
McGILL UNIVERSITY
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATH 270, Applied Linear Algebra 'Spring 2007
Date: Wednesday, April 25, 2007 Time: 9:00 am 12:00 pm
Examiner: Basak Giirel
Associate Examiner: Jim Loveys
INSTRUCTIONS
\
1. This exam has 11 pages
Linear algebra - Wikipedia, the free encyclopedia
3 of 5
http:/en.wikipedia.org/wiki/Linear_algebra
Some useful theorems
Every vector space has a basis.[3]
Any two bases of the same vector space have the same cardinality; equivalently, the dimension of a
Linear algebra - Wikipedia, the free encyclopedia
1 of 5
http:/en.wikipedia.org/wiki/Linear_algebra
Linear algebra
From Wikipedia, the free encyclopedia
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also c
McGill University
Department of Mathematics and Statistics
MATH 270 Applied Linear Algebra
Fall 2011
Midterm Solutions
1. (10 points) Consider the set R2 together with the following operations:
(x, y) + (x , y ) := (x + x , y + y ) i.e. standard vector ad
McGill University
Department of Mathematics and Statistics
MATH 270 Applied Linear Algebra
Fall 2011
Assignment 3: Solutions
1. Consider the map L : Mat(n n, R) R, L(A) = trace(A), where Mat(n n, R) is the
vector space of all n n matrices with real coecie
McGill University
Department of Mathematics and Statistics
MATH 270 Applied Linear Algebra
Fall 2011
Assignment 5: Solutions
1. Let A be an n n matrix with coecients in K.
(a) Prove that A is invertible i 0 is not an eigenvalue of A.
(b) Let A be invertib
McGill University
Department of Mathematics and Statistics
MATH 270 Applied Linear Algebra
Fall 2011
Assignment 4: Solutions
1
2
1. Compute the determinant of A = 2
7
5
3 7 9 8
4 8 11 9
2 1
2 2.
7 3
7 5
5 1
5 2
Remark: This problem has a simple and short
McGill University
Department of Mathematics and Statistics
MATH 270 Applied Linear Algebra
Fall 2011
Assignment 1: Solutions
1. Let z, w C. Prove that zw = z w.
Solution:
Let z = x1 + iy1 , w = x2 + iy2 . Then
zw = (x1 x2 y1 y2 ) + (x1 y2 + x2 y1 )i = (x1
McGill University
Department of Mathematics and Statistics
MATH270 Applied Linear Algebra
COURSE OUTLINE
Instructor:
Dr. Sidney Trudeau, Burnside Hall 1243
E-mail: trudeau@math.mcgill.ca
Office hours: MW 11:30-2:30 in either of BH1006A or BH911
Teaching A
Linear algebra - Wikipedia, the free encyclopedia
2 of 5
http:/en.wikipedia.org/wiki/Linear_algebra
When I went to Edinburgh as a young lecturer in 1922, I was surprised to find how different the curriculum
was from that at Oxford. It included topics such
Final exam of Applied Linear Algebra - Math 270
April 19, 2011
Problem 1
(2 pts.)
1
2
Find two dierent PLU decompositions for A =
2
1
.
SOLUTION:
1
A=
1
=
Problem 2
1
2
0
1
1
0
1
1/2
2
3
(1 pt.)
2
1
3/2
1
.
(1 pt.)
(2 pts.)
Let be the subspace of R3 dened