Week 13 Notes:
1) Least Squares Method
Aim: Given a system of inconsistent equations. Try to get the best possible solution out of
the constraints.
Example:
+ = 1
2 = 2
2 + = 3
is an inconsistent system. We want to find , , so that
+ =
a) 2 = is
Math2111 Introduction to Linear Algebra
Fall 2011
Final Examination (All Sections)
Name:
Student ID:
Lecture Section:
There are SEVEN questions in this final examination.
Answer all the questions.
You may write on both sides of the paper if necessary.
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MATH 2111 Matrix Algebra and Applications
Homework-1 : Due 02/20/2016 at 09:00pm HKT
Name: Jiachen LI
2. (1 pt)
x
y
5x 2y
4x
y
1. (2 pts)
For each system, determine whether it has a unique solution
(in this case, find the solution), infinitely many solu
MATH 2111 Matrix Algebra and Applications
Homework-4 : Due 04/09/2016 at 09:00pm HKT
Name: Jiachen LI
ACDE
1. (2 pts) Which of the following sets are subspaces of R3 ?
4. (3 pts) Let x, y, z be (non-zero) vectors and suppose w =
2y x + 3z.
x+
y.
If z = x
MATH 2111 Matrix Algebra and Applications
Homework-5 : Due 04/23/2016 at 09:00pm HKT
Name: Jiachen LI
1
? 3. S =
2
1
? 4. S =
2
4
8
,
is a basis for R2 .
5 19
24
Find the coordinate vector of x =
relative to the basis
48
B:
[x]B =
1. (1 pt) The set B =
MATH 2111 Matrix Algebra and Applications
Homework-2 : Due 03/05/2016 at 09:00pm HKT
Name: Jiachen LI
9
5
4
E. 2 , -9 , 11
6
-2
4
7
F.
-8
1. (2 pts) Do the columns of the matrix span R3 ?
? 1. A =
? 2. A =
? 3. A =
? 4. A =
-5 -5 5
7 7 -7
1 1 -1
18.06.01: Dimension
Lecturer: Barwick
Wednesday 03 February 2016
18.06.01: Dimension
What is dimension?
Line segments are 1-dimensional; heres one now:
Planar regions are 2-dimensional; heres one:
Finally, cubes are 3-dimensional:
Question. What have thes
Week 5 Notes:
1) Relation of Matrix Transformations and Linear Transformations
We noted that matrix transformations are linear transformations. One may ask, can linear
transformations be realized as matrix transformations. The answer is YES as we show bel
(MATH2111)[2015](f)final~=5jqrp^_30475.pdf downloaded by ibatra from http:/petergao.net/ustpastpaper/down.php?course=MATH2111&id=6 at 2016-09-10 22:08:59. Academic use within HKUST only.
Qn. 1 (20 marks) Choose a correct option for each question. No justi
MATH 2111 Matrix Algebra and Applications
Homework-2 : Due 10/08/2016 at 09:00pm HKT
Name: Ishaan BATRA
3
D.
,
-9
5
E. 4 ,
0
-2
F.
8
1. (2 pts) Do the columns of the matrix span R3 ?
? 1. A =
? 2. A =
? 3. A =
? 4. A =
2
2
7
5
-5
1
-9
2
1
6
9
8
8
MATH 2111 Matrix Algebra and Applications
Homework-1 : Due 09/24/2016 at 09:00pm HKT
Name: Ishaan BATRA
2. (1 pt)
x 4y
x + 3y
3x 5y
1. (2 pts)
For each system, determine whether it has a unique solution
(in this case, find the solution), infinitely many s
(MATH2111)[2011](f)final~=hneh5he^_82709.pdf downloaded by ibatra from http:/petergao.net/ustpastpaper/down.php?course=MATH2111&id=1 at 2016-09-10 22:08:51. Academic use within HKUST only.
Math2111 Introduction to Linear Algebra
Fall 2011
Final Examinati
(MATH2111)[2014](f)midterm~=w1sgpimcb^_65491.pdf downloaded by ibatra from http:/petergao.net/ustpastpaper/down.php?course=MATH2111&id=4 at 2016-09-10 22:08:56. Academic use within HKUST only.
Qn. 1 (20 marks) Choose a correct option for each question. No
MATH 2111 Matrix Algebra and Applications
Homework-6 : Due 05/09/2016 at 09:00pm HKT
Name: Jiachen LI
E. If x is orthogonal to every vector in a subspace W ,
then x is in W .
0
-3
1. (1 pt) Let x = -4 and y = 2 .
-1
-3
Find the dot product of x and y.
.
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MATH 2111 Matrix Algebra and Applications
Week 3 Tutorial
1. Consider a linear system with p variables and q equations.
(a) What are the sizes of the coefficient and augmented matrices?
(b) If p > q, is the system always consistent?
(c) If p < q, is the s
MATH 2111 Matrix Algebra and Applications
Week 6 Tutorial
x+y
xy
4
3
0
x + z . Find suitable x, y, z such that: (i) A is symmetric;
2x y + z
1
x+y
(ii) A is skew-symmetric.
1. Let A =
2. Let A be an m n matrix. Do we always have (i) AT A (ii) AAT being s
Week 10 Notes:
1) Coordinate Vectors
What is good about basis? It gives an efficient way to describe a space :
Theorem:
Suppose = , ,
is a basis of a vector space . Then every can be uniquely
written as
= + +
i.e. the values , , , are unique.
Examp
Week 13 Notes:
1) Orthogonal Matrices
We study the relationship between orthogonal matrices and orthonormal sets/bases.
Recall that an matrix is orthogonal if
Writing , then the , entry of is given by
If is an orthogonal matrix, then it says tha
Week 3 Notes:
1) Vector Equations
In this section, we relate system of linear equations with the study of vectors. Here is an example:
Example:
1
1
1
2
Is = 2 a l.c. of the vectors
= 1 , = 2 , = 1 ?
3
2
1
1
Solution: Do there exist , , such that:
1
Week 1 Notes:
0) Introduction
In short, linear algebra is a study of vectors. Visually, a vector is an arrow in a space (3-dimensional
vector space):
The 3-dimensions are left-right (-axis), forward-backward (-axis) and up-down (-axis). This vector
space
Week 11 Notes:
1) Diagonalizable Matrices
Definition:
A matrix is diagonalizable if there is an invertible matrix such that
=
0
0
=
Unfortunately our course does not cover the nicest perspectives on diagonalizable
matrices (keyword: change of bas
Week 12 Notes:
1) Inner Product of Vector Spaces
Prototype: The dot product of two vectors in :
Suppose , are two vectors in , then the dot product is defined as:
,
For instance, if 2 or 3, then the dot product in or gives
| cos !
Wher
Week 7 Notes (a):
1) Some Techniques in Computing Determinant
a) Expanding on other rows or columns
Theorem:
det can also be obtained by using cofactors on other rows/columns. i.e.
det
det
Example:
We can find determinant by expanding t
(MATH2111)[2012](f)midterm~=2qq1j^_80392.pdf downloaded by ibatra from http:/petergao.net/ustpastpaper/down.php?course=MATH2111&id=2 at 2016-09-10 22:08:52. Academic use within HKUST only.
HKUST
MATH2111 Introduction to Linear Algebra
Midterm Examination