Nail Ramadan
20568655
a
b
2. (10 marks) Let A =
b
b
b b b
a b b
where b 0 . Prove that A is diagonalizable.
b a b
b b a
To find the eigenvalues of A, we first compute the following determinant:
a
b
Review of linear algebra
Vectors
Vectors
A vector is often introduced in high school physics as a number and a direction.
Vectors can also be seen as abstract mathematical objects.
From a mathematical
Nail Ramadan
20568655
3. (10 marks) Let n n , and let L : n n be the linear mapping defined by
L ( x) = x 2
xn
n
2
n for all x n
a) Show that if y n , such that y 0 and y n = 0 , then y is an eigenvec
Nail Ramadan
20568655
Math 636 - Assignment 9 - Written Component
Due: Friday, July 15 at 4:00PM
Note: Your solutions should be uploaded to Crowdmark and 6 pages long.
2 3 1
1. (10 marks) Let A = 2 1
The mathematics of quantum mechanics
Postulate 4: Quantum operations
Postulate 4: Quantum operation
So far, weve learned how to calculate the measurement probability given the state of
a system. But h
Math 235
Assignment 5 Solutions
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1
2
be a subspace of R4 . Find a basis for the S .
1. Let S =
0
1
x1
1
x1
x2
2
S . Then
Multiple qubit systems
Postulate 5: Multiple quantum systems
Multiple quantum systems
So far, we have only investigated single quantum systems.
What if I want to describe a situation were I have multi
Nail Ramadan
20568655
Math 636 - Assignment 7 - Written Component
Due: Saturday, July 2 at 4:00PM
Note: Your solutions should be uploaded to Crowdmark and 7 pages long.
1. (5 marks) Let f ( x1 , x2 )
Math 674-7: Assignment 3 solutions
Throughout the questions, we will use the conventional notation:
1
1
1
0
1
1
|0i =
, |1i =
, |+i =
, |i =
0
1
2 1
2 1
1
| + ii =
2
1
i
1
| ii =
2
,
1
i
1.
Math 636 - Assignment 10 - Written Component
Due: Friday, July 22 at 4:00PM
1. Consider the function on R2 defined by
v1
w
, 1
= 2v1 w1 v1 w2 v2 w1 + v2 w2
v2
w2
(a) Prove that this function is an i
Math 636 - Assignment 8 - Written Component
Due: Friday, July 8 at 4:00PM
1 0
1 1 2
4 1 1 .
1. Find a basis for the four fundamental subspaces of A = 3 1
1 3 2 2 2
2. Prove of disprove the following s
Math 636 - Assignment 9 - Written Component
Due: Friday, July 15 at 4:00PM
2 3 1
1. Let A = 2 1 1.
2 3 5
(a) Find all of the eigenvalues of A and state their algebraic multiplicity.
(b) Find a basis f
Math 136
Assignment 4 Solutions
1. For each of the following systems of linear equations:
i) Write the augmented matrix.
ii) Row-reduce the augmented matrix into RREF.
iii) Find the general solution o
Math 136
Sample Midterm 2 Answers
NOTE: - Only answers are provided here (and some proofs). On the test you must provide
full and complete solutions to receive full marks.
1. Short Answer Problems
a)
Math 136
Sample Term Test 2 # 2 Answers
NOTE: - Only answers are provided here (and some proofs). On the test you must provide
full and complete solutions to receive full marks.
1. Short Answer Proble
Math 136
Additional Practice for Term Test 1 Solutions
1
5
1
1 , 1 , 1 .
1: Let B =
2
3
0
0
(a) Determine if ~v = 5 is in span B.
5
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Math 136
Assignment 8 Solutions
1. For each of
invertible.
1
(a) A = 2
2
the following matrices, find the inverse, or show that the matrix is not
2 4
2 3
3 5
[A | I] gives
1 0 0 1 2 2
2 4 1 0 0
2 3
Math 136
Assignment 10
1. Calculate the determinant of the following matrices.
1 10
2 4 0 0
7 7
1 2 2 9
(b) B =
(a) A =
2 2
3 6 0 3
3 3
1 1 0 0
Due: Wednesday, Mar 26th
7 9
7 7
6 2
4 1
sh is
ar s
Pre-Calculus MT022
Inverse Trigonometric
Functions
Objectives
Find
the Exact value of Sine Cosine
and Tangent Functions (p429)
Find the Approximate value of Sine
Cosine and Tangent Functions (p430)
Pre-Calculus MT022
Trigonometric Identities
Chapter 7
Objective
Recap
Use Algebra to Simplify Trigonometric
Identities
Establish Identities
Recap
S in
tan
Cos
Cos
Cot
S in
1
C o sec
s in
1
S
Pre-Calculus MT022
Exponential and Logarithmic
Functions
Objectives
Recap
Properties of Logarithm
Solving Logarithmic & Exponential
Equations
Write a logarithmic Expression as a
Sum or Difference
Pre-Calculus MT022
Exponential and Logarithmic
Functions
Objectives
Recap
Solving Logarithmic Equations
Properties of Logarithm
Write a logarithmic Expression as
a Sum or Difference of Logarithms
Chapter 7 Supplement: An
Application of FunctionsSound
What Is Sound?
Sound consists of vibrations of the air. In the air there are a large number of molecules moving around at about 1,000 mph, collid
APPENDIX B
Using the TI-84 Plus
Graphing Calculator
This appendix is intended to give you brief instructions and tips for some useful features that can help you explore some of the concepts in this bo