Course information Linear equations
10.004 Advanced Math 2
Cohort 1: Systems of Linear Equations
Term 2, 2015
1 / 22
Course information Linear equations
Introduction
Lecturer: james [email protected]
Carefully read the Syllabus on eDimension.
You will also
Basis for Rn Orthogonal basis Change of basis
10.004 Advanced Math 2
Cohort 11: Orthogonal Basis, Change of Basis
Term 2, 2015
1 / 26
Basis for Rn Orthogonal basis Change of basis
Basis for Rn
Consider a set of m vectors in Rn .
If m > n, then the vectors
Linear independence Matrix subspaces
10.004 Advanced Math 2
Cohort 10: Linear Independence and Matrix Subspaces
Term 2, 2015
1 / 30
Linear independence Matrix subspaces
Linear dependence
Denition: a set of vectors S = cfw_v 1 , v 2 , . . . , v k is calle
Vector spaces
10.004 Advanced Math 2
Cohort 9: Vector Spaces
Term 2, 2015
1 / 21
Vector spaces
Inner product space
Vector space
Denition: let V be a set with two operations, called addition and
scalar multiplication. If the following properties (axioms) h
Linear combination Dot product Cross product
10.004 Advanced Math 2
Cohort 8: Vectors and Geometry
Term 2, 2015
1 / 24
Linear combination Dot product Cross product
Unit vectors
Denition: a vector u is called a unit vector if its length is 1, that
is, if k
Determinant and properties Cofactor expansion
10.004 Advanced Math 2
Cohort 7: Determinants
Term 2, 2015
1 / 25
Determinant and properties Cofactor expansion
Properties
Determinant
Denition: the determinant of an n n matrix A is the signed
(or oriented) v
Elementary matrices
10.004 Advanced Math 2
Cohort 6: Matrix Inverse
Term 2, 2015
1 / 22
Elementary matrices
Elementary matrix
Denition: an elementary matrix is a matrix that can be obtained
by performing an elementary row operation on an identity matrix.
Matrix properties Inverse
10.004 Advanced Math 2
Cohort 5: Matrix Properties
Term 2, 2015
1 / 25
Matrix properties Inverse
Associativity
An important property of matrix multiplication is associativity.
Theorem
Matrix multiplication is associative, that is
Vectors Dot product
10.004 Advanced Math 2
Cohort 2: Vectors and the Dot Product
Term 2, 2015
1 / 20
Vectors Dot product
Vectors in R2
y
[3, 2]
A
[ 1, 1]
C
x
O
[ 3, 1]
B
!
!
a = OA = BC = [3, 2].
2 / 20
Vectors Dot product
Vector operations
Vectors can be
Echelon form General solution
10.004 Advanced Math 2
Cohort 4: Row Echelon Form
Term 2, 2015
1 / 25
Echelon form General solution
Elementary row operations
Let us take a more detailed look at the elimination steps, in terms
of what is happening to the (au
Linear transformations
10.004 Advanced Math 2
Cohort 12: Linear Transformations
Term 2, 2015
1 / 26
Linear transformations
Projections
Linear transformation
Denition: a function T : Rn ! Rm is called a linear
transformation if for all vectors u, v 2 Rn an
Multivariable Functions
10.004 Advanced Math 2
Cohort 15: Multivariable Functions
Term 2, 2015
1 / 23
Multivariable Functions
Level curves Partial derivatives
Graphs
Left: y = g(x) = x2 is a onevariable function. Its graph is a curve
in 2dimensional spa
Tangent planes Directional derivatives
10.004 Advanced Math 2
Cohort 16: Tangent Planes and Directional Derivatives
Term 2, 2015
1 / 23
Tangent planes Directional derivatives
Introduction
The idea of approximating a complicated function by a simpler
funct
PREFACE
New Syllabus Mathematics is a series of four books. These books follow the Mathematics Syllabus for
Secondary Schools, implemented from 2007 by the Ministry of Education, Singapore. The whole series
covers the complete syllabus for the SingaporeC
PREFACE
New Syllabus Mathematics Workbook (Express) is written in line with the new SingaporeCambridge GCE '0' Level Examination and the new initiatives of the Ministry of Education.
The workbook consists of exercises which prepare students for their exam
PREFACE
New Syllabus Mathematics Workbook 4 (Express) is written in line with the new SingaporeCambridge GCE '0' Level Examination and the new initiatives of the Ministry of Education.
The workbook consists of exercises which prepare students for their ex

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PREFACE
New Syllabus Mathematics Workbook 4 (Express) is written in line with the new SingaporeCambridge GCE '0' Level Examination and the new initiatives of the Ministry of Education.
The workbook consists of exercises which pr
6
Algebraii
6.1
Algebraic Fractions
6.2
Solving Linear Equations involving Algebraic Fractions
6.3
Changing the Subject of a Formula
6.4
Solving Simultaneous Linear Equations in Two Unknowns
Revision Exercise
7
Arithmetic I
1.1
Numbers and the Four Operat
PREFACE
New Syllabus Mathematics Workbook (Express) is written in line with the new SingaporeCambridge GCE '0' Level Examination and the new initiatives of the Ministry of Education.
The workbook consists of exercises which prepare students for their exam
09. Data Analysis
=/


x f
Vocabulary
Mean
Median
Mode
Class interval
midvalue
Standard deviation
Quartile
Percentile
In the previous section, we have learnt several ways to display data such as pictograms, bar
charts and lines graphs. Besides, we sti
Min & max
10.004 Advanced Math 2
Cohort 18: Minimum & Maximum
Term 2, 2015
1 / 25
Min & max
Hessian
Max and min for single variable functions
For a twicedierentiable, single variable function f , a stationary
point is a point x0 where f 0 (x0 ) = 0. The
Chain rule Implicit dierentiation
10.004 Advanced Math 2
Cohort 17: Chain Rule and Implicit Functions
Term 2, 2015
1 / 19
Chain rule Implicit dierentiation
Motivation
For a function of one variable y = f (x) where x = g(t) (that is,
y = f (g(t), the chain
Course information Linear equations
10.004 Advanced Math 2
Cohort 1: Systems of Linear Equations
Term 2, 2015
1 / 17
Course information Linear equations
Simple example
Consider the linear system of equations:
2x + y =
x
3y =
8
3
Solve for x and y.
There i
Matrix multiplication Elimination
10.004 Advanced Math 2
Cohort 3: Matrices
Term 2, 2015
1 / 22
Matrix multiplication Elimination
Matrix multiplication
Matrices can only be multiplied if the number of columns of the
rst matrix is equal to the number of ro
Matrix properties Inverse
10.004 Advanced Math 2
Cohort 5: Matrix Properties
Term 2, 2015
1 / 13
Matrix properties Inverse
Properties of matrix multiplication
In Cohort 3, Activity 2, we looked at some basic properties of
matrix addition, scalar multiplic
Echelon form General solution
10.004 Advanced Math 2
Cohort 4: Row Echelon Form
Term 2, 2015
1 / 13
Echelon form General solution
Elementary row operations
Let us take a more detailed look at the elimination steps, in terms
of what is happening to the (au
Matrix multiplication Elimination
10.004 Advanced Math 2
Cohort 3: Matrices
Term 2, 2015
1 / 15
Matrix multiplication Elimination
Matrices
Denition: a matrix is a rectangular array of numbers. If a matrix
has m rows and n columns, we say that the size of
Vectors Dot product
10.004 Advanced Math 2
Cohort 2: Vectors and the Dot Product
Term 2, 2015
1 / 14
Vectors Dot product
Vectors
!
In R2 and R3 , a vector AB is a directed line segment that
corresponds to a displacement from point A to point B.
You have s
Course information Linear equations
10.004 Advanced Math 2
Cohort 1: Systems of Linear Equations
Term 2, 2015
1 / 12
Course information Linear equations
Introduction
Lecturer: james [email protected]
Carefully read the Syllabus on eDimension.
You will also
10.004 Advanced Math 2, Homework Set 5
Due 6pm, Monday 9 November 2015
Please show all working, and clearly indicate your name, cohort and student ID.
1. Let A be a square matrix.
(a) Show that A and AT have the same eigenvalues. Do they necessarily have