MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
Gaussian Elimination (1.2; pg. 11) cont.
Recall: Last day, we introduced Gaussian and Gauss-Jordan Elimination for rowreducing a matrix. Lets get some more practice at this.
More Examples: Lets do some more
MATH1850 Midterm 1
October, 2011
2
1. (3 marks each; total 12 marks) Answer each of the following questions in the space
provided.
a) Solve the linear system associated with the given augmented matrix.
1 5 2 4 1
0 1 0 0 2
0 0 0 1 6
5 8
4 3
b) Given A
MATH 1850 Linear Algebra for Engineers Fall 09
Instructor: Lectures:
Mihai Beligan 40311 Wed 12:40pm-02:00pm UB2080 Thu 03:40pm-05:00pm UA1350 40312 Mon 11:10am-12:30pm UA1350 Thu 12:40pm-02:00pm UA1350
Office: UA4042 Office hours will be posted later e-m
Linear Algebra Midterm 1- v6-Solutions
MATH1850 Midterm 1
October, 2010
1. (3 marks each; total 6 marks) Solve the linear systems associated with the
following augmented matrices. Hint: They are already either in reduced row-echelon or
row-echelon form fo
Linear Algebra Midterm 1- V1
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Linear Algebra Midterm
1. (7 marks total)
(a) (4 marks) Find all unit (i.e. of length equal to l) vectors in R3 orthogonal
to the vectors u = (1,0, l) and V = (1,2,0).
START by Hunt, )_¢.: (5.x! (5m; 9m; _L 33!, an
i. 5 k
>_<= \ o ~l =(-1,-\,-23.
\ -2 o
X l
Mon) 1 \5 war A up'
Linear Algebra Midterm 1- v2-Solutions
MATH1850 Midterm 1
October, 2010
1. (3 marks each; total 6 marks) Solve the linear systems associated with the
following augmented matrices. Hint: They are already either in reduced row-echelon or
row-echelon form fo
Syllabus for Linear Algebra for Engineers,
Fall 2012
Course Information
Course number
MATH1850U-001, MATH1850U-002, MATH1850U-003
Course date
Thursday, September 6, 2012 through Wednesday, December 5, 2012
Lectures
Section 001: UP1500
Tues 12:40 2:00 pm
T
Midterm II V2- Solutions
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Midterm II V2- Solutions
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Midterm II V2- Solutions
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Midterm I
Linear Algebra Midterm 2- v7-Solutions
MATH1850 Midterm 2
November, 2010
2
1. (6 marks) Find the angle (in degrees) between the two vectors (4, 1, -2) and
(5,-3,1). Hint:
u v u v cos( )
2. (2 marks each; 4 marks total) Answer each question in the space pr
MATH1850U: Chapter 5 cont.; 6
1
EIGENVALUES AND EIGENVECTORS
Diagonalization (Section 5.2) cont.
Recall: Last day, we introduced the concept of diagonalizing a matrix.
Motivation: Diagonalization leads to an efficient way of computing large powers of a
k
MATH1850U: Chapter 6 cont
1
INNER PRODUCT SPACES cont
Inner Products (Section 6.1) cont
Recall: Last day, we introduced the concept of a weighted inner product, and how to
find distance and norm using this inner product.
Definition: If V is an inner produ
MATH1850U: Chapter 4 cont.
GENERAL VECTOR SPACES cont.
Properties of Matrix Transformations (Section 4.10)
Recall: Last day, we said that the standard matrix for a transformation can be found
using T T (e1 ) | T (e 2 ) | | T (e n ) .
Example: Find the sta
MATH1850U: Chapter 5
1
EIGENVALUES AND EIGENVECTORS
Eigenvalues and Eigenvectors (Section 5.1)
Definition: If A is an n n matrix, then a nonzero vector x in Rn is called an eigenvector
of A if Ax is a scalar multiple of x; that is
Ax x
for some scalar . T
MATH 2050U Term Test 1 Page 2 of 7
1. [5 marks] Indicate whether the following statements are always True or sometimes False.
(a) If A, B are invertible matrices of same size, then AB : BA
El True Er False
(b) The inverse of an elementary matrix is an
MATH 1850/ 2050U Terrn Test 2 Page 2 of 7
1. For each of the following, check m ALL boxes for the statements that are True.
PLEASE NOTE: none, or more than one answer per subquestion may be selected.
Each incorrect selection (an incorrect selection consis
Diu'v. scurrle
MATH 1850 / 2050U Term Test 2 Page 2 of 7
1. [5 marks] Indicate Whether the following statements are always True or sometimes False.
(a) If A is a 3 X 5 matrix, then its rank is at most 3
El True El False
(b) The plane 3x y @ 0 passes t
IN-TUTORIAL ASSIGNMENT #3
(to be done in tutorial the week of Oct 3 7)
INSTRUCTIONS: Refer to the syllabus for full details on how youll be graded, and how to get
feedback on your work.
Before Tutorial: Read section 1.9 of the text (Applications of Linear
IN-TUTORIAL ASSIGNMENT #4
(to be done in tutorial the week of Oct 31 Nov 4)
INSTRUCTIONS: Refer to the syllabus for full details on how youll be graded, and how to get
feedback on your work.
Before Tutorial: Read pg. 208-210 of the text (starting on pg. 2
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Row Space, Column Space, and Null Space (Section 4.7) cont.
a11
a
21
A
matrix
a m1
a12
a1n
a 22 a 2 n
m
n
Definition: For an
, the vectors in Rn formed
a m 2 a mn
from the rows of A are call
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Linear Independence (Section 4.3) cont.
Recall: Last day, we introduced the concept of linear dependence and independence.
Example: We mentioned that the standard unit vectors in R n , e1 , e 2 , ,
MATH1850U: Chapter 1
1
LINEAR SYSTEMS
Application Balancing Chemical Equations: Write a balanced equation for the given
chemical reaction: CO2 + H2O C6H12O6 + O2 (photosynthesis)
Application Population Migration: A country is divided into 3 demographic re