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
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Matrix Transformations from Rn to Rm (Section 4.9)
NOTE: Some of this material is also covered in section 1.8 of the text.
Recall: You are already familiar with functions from Rn to R; this is a rul
MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
Elementary Matrices and a Method for Finding A-1 (Section 1.5)
Definition: An n n matrix is called an elementary matrix if it can be obtained from
the n n identity matrix by performing a single elementary r
MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
Matrices and Matrix Operations (1.3; pg. 25) cont
Last day, we learned matrix multiplicationlets get a bit more practice!
Example:
Example: Find the 2nd column in the product AB.
product AB.
Exercise: Find
MATH1850U: Chapter 5 cont
1
EIGENVALUES AND EIGENVECTORS cont
Eigenvalues and Eigenvectors (Section 5.1) cont
Recall: Last day, we introduced the concept of eigenvalues and eigenvector.
Application: Markov Chains
Recall: On assignment #6, you had the oppo
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 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 4 cont.
1
GENERAL VECTOR SPACES cont.
Dimension (Section 4.5)
Definition: A nonzero vector space V is called finite-dimensional if it contains a finite
set of vectors v1 , v 2 , v n that form a basis. If no such set exists, V is called
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
MATH1850U: Chapter 2 cont.
1
DETERMINANTS cont.
Evaluating Determinants by Row Reduction (Section 2.2)
Recall: Last day, we introduced the method of cofactor expansion for finding
determinants. Today, we will learn to evaluate determinants by row reductio
MATH1850U: Chapter 1 cont.; 2
1
LINEAR SYSTEMS cont
Applications of Linear Systems (Section 1.9)
Recall: Weve spent the past few weeks studying techniques for solving a system of
equations. So lets talk a bit more about the applications of this!
Applicati
MATH1850U: Chapter 2 cont.; 3
1
DETERMINANTS cont.
Properties of Determinants; Cramers Rule (Section 2.3) cont.
Recall: Last class, we began studying several properties of determinants.
Theorem (Cramers Rule): If Ax b is a system of n equations in n unkno
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 , e1 , e 2 , e n
MATH1850U: Chapter 4
1
GENERAL VECTOR SPACES
Recall: In Chapter 3, we saw n-space or Rn. All together, the following 3 things make
up n-space:
1. The objects
2. Rule for addition: a rule for associating with each pair of objects u and v an
object u v , ca
MATH1850U: Chapter 3 cont.
1
EUCLIDEAN VECTOR SPACES cont.
Norm, Dot Product, and Distance in Rn (Section 3.2) cont.
Recall: Last day, we introduced the concept of dot product.lets examine some
properties.
Theorem (Properties of the Euclidean Inner Produc
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Subspaces (Section 4.2) cont.
Recall: Last day, we introduced the concept of subspace and linear combination.
Definition: If S v1 , v 2 , v r is a set of vectors in a vector space V, then the subspa
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Subspaces (Section 4.2)
Definition: A subset W of V is called a subspace of V if W is itself a vector space under
the addition and scalar multiplication defined on V.
Theorem (Subspaces): If W is a
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Row Space, Column Space, and Null Space (Section 4.7) cont.
Recall: Last day, we introduced the concept of row, column, and null space.
Theorem: Elementary row operations do not change the row space
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
1. (10 marks total) Solve the following linear system through Gaussian Elimination
or GaussJordan Elimination.
2.E1 33:2 + {1:3 I 7
{1?1 + 23:2 {E3 = 5
3.E1 63:2 = 3
1 =5 \ 7
I 'L -L 5
5 L 3
Rage-'- l 'L l S
_9 L -'b I 7
($4205.) i L 'L -\
L m | L -l S
IL
8. (5 marks) True/ False. Indicate Whether the following statements are always True or
sometimes False.
(a) Elementary row operations do not change the nullspace of a matrix. T
(b) The planes 3y + Z = 0 and 5.1: + 3y Z + 3 = 0 are perpendicular. (r
V\.=(.
Linear Algebra Midterm 1- V1
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Linear Algebra Midterm
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
OBJECTIVES:
Section 5.2 By the end of this section, you will be able:
to write down that a matrix A is diagonalizable if there is an (invertible) matrix P such that
D = P 1 AP is a diagonal matrix
to find the matrix P that diagonalizes a 3 3 matrix A
OBJECTIVES:
Section 6.1 By the end of this section, you will be able:
to state that an Inner Product Space is a Vector Space together with an inner product
to state that the inner product of any two vectors is a real number
to write down the Euclidean
OBJECTIVES:
Section 6.3 By the end of this section, you will be able:
to state that a set of vectors is an orthogonal set if all pairs of vectors in the set are orthogonal
to state that an orthogonal set of vectors in which all vectors have norm 1 is c
OBJECTIVES:
Section 7.1-2 By the end of these sections, you will be able:
to state that an orthogonal matrix is a matrix whose inverse is its tranpose
to state that a matrix A is orthogonally diagonalizable when there is an orthogonal matrix P
that dia
OBJECTIVES:
Section 5.1 By the end of this section, you will be able:
to write down the equation that defines the eigenvalues and eigenvectors of a matrix
to state that if x is an eigenvector of a matrix A with a real eigenvalue t, then the vector Ax
i