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SINGAPORE UNIVERSITY OF
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10.001: Advanced Math 1
Term 1, 2015
Homework 01
Due Date: May 23, 2016 - 17:00 (5pm)
1. Labelpoints A, B, C, D, E and F on the graph of y = f (9:) in the ﬁgure s
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 twice-dierentiable, 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
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
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 one-variable function. Its graph is a curve
in 2-dimensional spa
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
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 wan@sutd.edu.sg
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
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
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
10.004 Advanced Math 2, Homework Set 4
Due 6pm, Monday 2 November 2015
Please show all working, and clearly indicate your name, cohort and student ID.
1. (a) Find the projection matrix, P , for projecting a vector onto a line in the direction of
T
a = 1 2
10.004 Advanced Math 2, Homework Set 3
Due 6pm, 19 October 2015
Please show all working, and clearly indicate your name, cohort and student ID.
1. (a) Compute det(A) using cofactor expansion along the second row.
k
k
3
1
A = 0 k + 1
k
8 k 1
(b) Check you
10.004 Advanced Math 2, Homework Set 2
Solutions
1.
2
1
RRT = 40
0
2
1
= 40
0
0
cos
sin
32
0
1
sin 5 40
cos
0
3
0
sin 5
cos
0
cos
sin
3
0
cos sin + cos sin 5
( sin )2 + cos2
0
cos2 + sin2
cos sin + cos sin
= I3 ,
since cos2 + sin2 = 1.
We can sim
Min & max
10.004 Advanced Math 2
Cohort 18: Minimum & Maximum
Term 2, 2015
1 / 13
Min & max
Hessian
Max and min for single variable functions
For a twice-dierentiable, single variable function f , a stationary
point is a point x0 where f 0 (x0 ) = 0. The
Fourier Tangent planes Directional derivatives
10.004 Advanced Math 2
Cohort 16: Tangent Planes and Directional Derivatives
Term 2, 2015
1 / 24
Fourier Tangent planes Directional derivatives
Fourier series
In Cohort 11 Activity 3, we saw that the set of f
Chain rule Implicit dierentiation
10.004 Advanced Math 2
Cohort 17: Chain Rule and Implicit Functions
Term 2, 2015
1 / 10
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
Diagonalization
10.004 Advanced Math 2
Cohort 14: Diagonalization
Term 2, 2015
1 / 11
Diagonalization
Introduction
In many applications, we are required to e ciently compute An for
a square matrix A.
The trick is to write, if possible,
A = P DP
1
,
where
Eigenvalues and eigenvectors
10.004 Advanced Math 2
Cohort 13: Eigenvalues and Eigenvectors
Term 2, 2015
1 / 15
Eigenvalues and eigenvectors
Introduction
Thinking of a matrix as the linear transformation it represents
allows us to understand it better.
Fo
Linear transformations
10.004 Advanced Math 2
Cohort 12: Linear Transformations
Term 2, 2015
1 / 13
Linear transformations
Introduction
We now look at an application of matrices to linear
transformations, that is, familiar geometric transformations such a
Basis for Rn Orthogonal basis Change of basis
10.004 Advanced Math 2
Cohort 11: Orthogonal Basis, Change of Basis
Term 2, 2015
1 / 13
Basis for Rn Orthogonal basis Change of basis
Introduction
Recall that for a vector space V , a basis is a set of linearl