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CIRCLE THE NUMBERS OF YOUR LECTURE AND LAB SECTIONS:
001
002
003
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005
MWF 8:30
MWF 10:30
Wed 9:30 Youlong Yan
Thu 2:30 Allen OHara
Thu 11:30 Jason Haradyn
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40
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UWO ID number:
CIRCLE THE NUMBERS OF YOUR LECTURE SECTION AND LAB SECTION IN THE TABLES
BELOW:
001
002
003
004
005
MWF 8:30
MWF 10:30
Wed 9:30 Youlong Yan
Thu 2:30 Allen
Student Solution Manual for
Introduction to Linear Algebra
Geza Schay and Dennis Wortman
1.1.1. P R = r p, P Q = q p, and QP = p q.
QC = 1 QP = 1 p 1 q, P C = 1 P Q = 1 q 1 p, and OC = 1 r = 1 p+ 1 q.
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1.1.3. p + q = (2, 3, 1) + (1, 2, 2)
LINEAR ALGEBRA 1600 SUMMER 2010 MIDTERM MAY 18, 2010
First Name
Last Name
Student Number
Please read the instructions given below.
PRINT your last name, rst name, and student number above.
This exam is due on Tuesday, May 25th at 7:00 PM in our usual cl
Linear Algebra 1600a Midterm
7:00-10:00 pm
October 30, 2009
(16 pts) 1. For each of the following, circle T if the statement is always true and circle F if it can be false.
If you are unsure, leave blank. Wrong answers will receive 2 marks.
We justify the
Linear Algebra 040b Midterm Examination Saturday, March 11, 2006
1. [2 marks ] Find the sum of the vectors (3, 1, 2, 4) and (4, 8, 1, 1).
2. [2 marks ] Let be the angle between the vectors (1, 2, 3) and (3, 3, 1). Find cos .
3. [2 marks ] Find the value o
Linear Algebra 040a Midterm Examination Friday, November 4, 2005
1. [2 marks ] Find the dot product of the vectors (1, 0, 0, 3, 2) and (2, 3, 0, 2, 2).
2. [2 marks ] Let be the angle between the vectors (0, 1, 0) and (2, 1, 2). Find cos .
3. [2 marks ] Fi
Linear Algebra 1600a Midterm
Last Name
7:00-10:00 pm
October 30, 2009
1 First Name Student ID CIRCLE LECTURE AND LAB SECTIONS: LECTURE: 001 MWF 8:30 LAB: 003 W 9:30 002 MWF 10:30 005 Th 11:30 006 W 3:30 19
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70
This exam has 11 pr
Polar form of a Complex number
i Polar Example: Let z = 1 + i. Then i form of z = |z |e . For z = 1 + i, |z | = 12 + 12 = 2 and = /4. So z = 2e 4
Multiplication: Let z1 = |z1 |ei1 , z2 = |z2 |ei2 then z1 z2 = |z1 |z2 |ei(1 +2 ) = |z1 |z2 |(cos (1 + 2 ) +
Section 2.2: Evaluating Determinants by Row Reduction We will look at the relationship between the determinants of row-equivalent matrices. Recall we have 3 elementary matrix operations 1. Interchange two rows 2. Muptiply a row bya nonzero constant 3. Add
Section 5.3 Complex Vector Spaces & Appendix B Denition: A complex number is an ordered pair of real numbers, denoted either by (a, b) or by a + bi where i = 1. Usual notation is z = a + bi, a=Re(z) is called the real part of z, b=Im(z) is called the imag
Section 4.3 Linear Independence Linear indepencency of vectors will be used to dene basis of a vector space that we will see in section 4.4 and to determine the dimension of a space that we will see in section 4.5. Denition: A non-empty set of vectors S =
Section 5.2 Diagonalization Denition: If A and B are square matrices, then we say that B is similar to A if there is an invertible matrix P such that B = P 1 AP. Facts: 1. A and B have the same determinant 2. A is invertible if and only if B is invertible
Theorem 1.7.1: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product of lower triangular matrices is lower triangular, a
Denition: A matrix transformation T : Rn Rm is said to be onto if evey vector in Rm is the image of at least one vector in Rn . Theorem 8.2.2: If T is a matrix transformation, T : Rn Rn , then the following are equivalent (a) T is one-to-one (b) T is onto
Lets see some more examples of nding standard matrix of a matrix transformation Example: Find the standard matrix of the given operators 1. T : R3 R3 , reection through the xy-plane 2. T : R3 R3 , reection through the plane x=z 3. T : R3 R3 , Dilation wit
Theorem 1.5.1 suggests that reducing a matrix A to (reduced) row echelon form is tha same as multiplying A from left by the appropriate elementary matrices. Hence if B is a matrix obtained from a matrix A by performing a nite sequence of elementary row o
Back to matrix multiplication: Recall for matrix addition we have zero matrix with A 0 = 0 + A = A for any matrix A. + 1 0 . 0 0 1 . . . 0 We have a similar element called identity matrix I = . . . . . such that AI = A. . . . . 0 0 . 1 Size of I must be a
Properties of Matrix Transformations Theorem 4.9.1: For every matrix A the matrix transformation TA : Rn Rm has the following properties for all vectors u and v in Rn and for every scalar k: (a) TA (0) = 0 (b) TA (ku) = kTA (u) (Homogeneity property) (c)