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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
b
b
b
a
b
b
b
b
a
b
b
b
b
a
det( A I ) = A I =
Accordi
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 perspective, a vector is just a way of stacking number
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 eigenvector of L. What is its eigenvalue?
Operating with L on v
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 1 .
2 3 5
(a) Find all of the eigenvalues of A and st
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 how do we create that state?
In other words, is it possi
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, we have 0 = x2 = x1 2x2 + x4 .
Solution: Let ~x =
x3
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 multiple quantum systems that
interact with each other?
How
Math 674-7: Assignment 1
1. [20 points, 2 points each] Calculate the following and write your answer in the
form a + bi.
(a) (1 3i) (4 + 7i)
(b) (1 + 9i)(1 + i)
(c) 1 + 2i
(d) 1 + 2i
(e) |1 + 2i|
1
(f)
(1 + 2i)
(g) ei + ei
(h) ei ei
(i) ei 4 ei 2
ei 3
(j)
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 ) (2 x2 , x1 3x2 , 2 x2 ) . Prove that f is linear and th
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. Without going too much in the details of the physics, s
Math 674-7: Assignment 5
1. [25 points, 5 points each] Find the explicit representation of:
(a) | + ii |0i
(b) |+i | ii
(c)
!
2ei 4
1
|0i + |1i
5
5
!
1
2ei 4
|0i + |1i
5
5
(d) H X
(e) Z Y
2. [20 points, 4 points each] We have been using the inner produ
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 inner product on R2 .
(b) Show that the standard basis v
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 statement: There exists a matrix A M33 (R) such that Nul
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 for the eigenspace of each eigenvalue of A.
(c) Determin
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 of the system or explain why the system is inconsistent.
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) State the definition of a basis of a subspace S of Rn .
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 Problems
a) Let B = cfw_~v1 , . . . , ~vn be a non-empty sub
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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Solution: We need to determine if there exists c1 , c
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 0 1 0 0 1 0 4 13 11
3 5 0 0 1
0 0 1 2 7 6
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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
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2
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)
Use Properties of Inverse Functions
to Find Exact valu
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 e c
Cos( ) Cos
cos
Sec( ) Sec
1
C o t
T a n
sin( ) 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 of Logarithms
Exercise
Pages 299 to 302
Evaluate Expo
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
Exercise
Pages 283 to 298
Evaluate Exponential Functi
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, colliding into each other. The collision of
the air molecules
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 book.
Animator
We can use the Animator feature to draw an