MATH1850U: Chapter 1
1
LINEAR SYSTEMS
Introduction to Systems of Linear Equations (1.1; pg.2)
Definition: A linear equation in the n variables, x1 , x2 , xn is defined to be an
equation that can be written in the form:
a1 x1 a 2 x2 a n xn b
where a1 , a 2

MATH1850U: Chapter 3 cont.
1
EUCLIDEAN VECTOR SPACES cont.
Norm, Dot Product, and Distance in Rn (3.2; pg. 130)
Definition: We define the Euclidean norm (or Euclidean length) of a vector
u (u1 , u 2 , u n ) in R n by
2
2
| u | (u u)1 2 u12 u 2 u n
Example

MATH1850U: Chapter 3 cont.; 4
1
EUCLIDEAN VECTOR SPACES cont.
The Geometry of Linear Systems (Section 3.4; pg. 152)
Recall: A line in R 2 can be specified by giving its slope and one of its points.
Similarly, one can specify a plane in R 3 by giving its i

MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Real Vector Spaces (4.1; pg. 171) cont.
Recall: Last day, we introduced the definition of a vector space. Lets do some more
examples.
Example: Let V be the set of real-valued functions defined on th

MATH1850U: Chapter 2 cont.; 3
1
DETERMINANTS cont.
Properties of Determinants; Cramers Rule (2.3; pg. 106) 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 unkn

MATH1850U: Chapter 2 cont.
1
DETERMINANTS cont.
Evaluating Determinants by Row Reduction (2.2; pg. 100)
Recall: Last day, we introduced the method of cofactor expansion for finding
determinants. Today, we will learn to evaluate determinants by row reducti

MATH1850U: Chapter 1 cont.
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.
Example (Gauss-Jordan elimination)

MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
Inverses and Algebraic Properties of Matrices (1.4; pg. 38)
Recall: Last day, we learned how to multiply matrices.
Caution: There is no Commutative Law for matrix multiplication! In other words, it is
NOT n

MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
More on Linear Systems and Invertible Matrices (1.6; pg. 60)
Recall: Last day, we learned about the concept of inverting a matrix.
Theorem: Every system of linear equations has either no solution, exactly o

MATH1850U: Chapter 1 cont.; 2
LINEAR SYSTEMS cont
Diagonal, Triangular, and Symmetric Matrices (1.7; pg. 66)
Recall: Weve spent a lot of time talking about how to work with matrices. Now lets
introduce a few special matrices.
Definition: A square matrix i

MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Subspaces (4.2; pg. 179) cont.
Recall: Last day, we introduced the concept of a subspace:
If u and v are vectors in W, then u v is in W
If k is any scalar and u is any vector in W, then ku is in W

MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Linear Independence (4.3; pg. 190) cont.
Recall: Last class, we introduced the notion of linear dependence and independence.
Example: Determine whether the
linearly independent set.
P2 vectors given

MATH1850U: Chapter 5 cont.; 6
EIGENVALUES AND EIGENVECTORS
Diagonalization (Section 5.2; pg. 305) cont.
Recall: Last day, we introduced the concept of diagonalizing a matrix.
Theorem: If A is an n n matrix, then the following are equivalent:
a) A is diago

MATH1850U: Chapter 6 cont
1
INNER PRODUCT SPACES cont
Inner Products (Section 6.1; pg. 335) 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 in

MATH1850U: Chapter 6 cont.
1
INNER PRODUCT SPACES cont.
Gram-Schmidt Process; QR-Decomposition (6.3; pg. 352)
Definition: A set of vectors in an inner product space is called an orthogonal set if all
pairs of distinct vectors in the set are orthogonal. An

MATH1850U: Chapter 5 cont
1
EIGENVALUES AND EIGENVECTORS cont
Eigenvalues and Eigenvectors (5.1; pg. 295) cont
Recall: Last day, we introduced the concept of eigenvalues and eigenvectors.
3 6
Example: Find the eigenvalues and bases for the eigenspaces of

MATH1850U: Chapter 5
1
EIGENVALUES AND EIGENVECTORS
Eigenvalues and Eigenvectors (5.1; pg. 295)
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 .

MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Dimension (4.5; pg. 209)
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 4 cont.
1
GENERAL VECTOR SPACES cont.
Row Space, Column Space, and Null Space (4.7; pg. 225) cont.
Recall: Last day, we introduced the concept of row, column, and null space.
Theorem: Elementary row operations do not change the row spac

MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Matrix Transformations from Rn to Rm (4.9; pg. 247)
Recall: You are already familiar with functions from Rn to R; this is a rule that associates
with each element in a set A one and only one element

MATH1850U: Chapter 4 cont.
GENERAL VECTOR SPACES cont.
Properties of Matrix Transformations (4.10; pg. 263)
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 st