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
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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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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)
Relation of row space and column space of A, to Ax=b Theorem 4.7.1: A system of linear equations Ax=b is consistent if and only if b is in the column space of A. x1 . Idea: Consider Ax = b where x = . , A = c1 . . . cn . c1 , . . . , cn denotes the . xn c
Chapter 4, General Vector Spaces Section 4.1, Real Vector Spaces In this chapter we will call objects that satisfy a set of axioms as vectors. This can be thought as generalizing the idea of vectors to a class of objects. Vector space axioms: Denition: Le
Section 4.6 Change of Basis Let B = cfw_v1 , v2 , . . . , vn be a basis for a nite dimensional vector space V. Let v V then we can express v as v = c1 v1 + c2 v2 + . . . + cn vn . Recall the coordinate vector of v, which was denoted as (v )B = (c1 , c2 ,
To determine the direction of u v we have the right hand rule; Fingers point the rst vector, Palm points the second vector, tumb gives the direction of the cross product.
Theorem 3.5.1 (Relationship involving Cross Product and Dot Product:) If u,v and w a