AMS 10/10A Mathematical Methods for Engineers I
Review for the Midterm Exam
The scope of the midterm exam is up to and includes Section 2.3 in the textbook (Lectures 1-9).
Let us highlight some of the important items.
Complex numbers
a + bi
Complex conjugate:
Arithmetics:
Example:
a1 + b1 i
a2 + b2 i
How to calculate this division?
The polar form
r ( cos + sin i )
The absolute value,
The argument,
Conversion between a + b i
r ( cos + sin i )
and
How to calculate the absolute value given the Cartesian form?
How to calculate the argument given the Cartesian form?
The exponential form
r e i = r ( cos + sin i )
Arithmetics in the exponential form:
r1 e1 i
Example:
r2 e2 i
How to calculate this division?
i
Example: 2 e 3 11
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AMS 10 Mathematical Methods for Engineers I
11
i
How to write out the exponential form of 2 e 3 ?
11
i
How to write out the Cartesian form of 2 e 3 ?
Roots of polynomials
The n-th roots of a complex number (there are n of them)
Example: The 7-th roots of (-2 + 2i)
How to write out ALL of the 7-th roots of (-2 + 2i)?
Linear systems, vectors, matrices
Elementary row operations
What are the 3 kinds of elementary row operations?
Row reduction to echelon form
Row reduction to reduced echelon form
What is the definition of echelon form?
What is the definition of reduced echelon form?
Pivot positions
What is a pivot position?
Pivot columns
What is a pivot column?
Basic variables
How do we identify basic variables?
Free variables
How do we identify free variables?
Solution set in parametric form
How to find the solution set of A x = 0 ?
How to find the solution set of A x = b ?
Question 1:
Is A x = b consistent for a particular given b ?
Method:
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AMS 10 Mathematical Methods for Engineers I
To answer this question, we look at if the rightmost column of the augmented matrix A b is a pivot column.
(Theorem 2 of Chapter 1)
Question 1B:
How to find pivot positions of the augmented matrix A b ?
Method:
Row reduction to an echelon form.
Question 2:
Suppose A x = b is consistent for the given b .
Does A x = b have a unique solution or infinitely many solutions?
Method:
To answer this question, we look at the number of free variables.
(Theorem 2 of Chapter 1)
Question 2B:
How to identify basic variables and free variables of A x = b ?
Method:
Identify pivot columns and …
Question 2C:
Suppose A x = b is consistent for the given b .
How to write out the solution set of A x = b ?
Method:
Row reduction to reduced echelon form;
Identify basic variables and free variables;
Express basic variables in terms of free variables.
Question 3:
Is b a linear combination of a1 , …, an ?
Method:
This question is equivalent to Question 1 with A = [ a1 an ] .
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AMS 10 Mathematical Methods for Engineers I
So we look at the pivot positions of the augmented matrix A b .
(Theorem 3 and Theorem 2 of Chapter 1)
Question 4:
Is b in span {a1 , …, an } ?
Method:
This question is equivalent to Question 1 with A = [ a1 an ] .
So we look at the pivot positions of the augmented matrix A b .
(Theorem 3 and Theorem 2 of Chapter 1)
Question 5:
Suppose matrix A is m n.
Is A x = b consistent for every b in m ?
Method:
To answer this question, we look at if there is a pivot position in every row of matrix A.
IMPORTANT: here we look at the pivot positions of matrix A, not the augmented
matrix A b .
(Theorem 4 of Chapter 1)
Question 5B:
Suppose matrix A is 11 9.
Is it possible that matrix A has a pivot position in every row?
Method:
Check the number of columns.
Can a column have more than 1 pivot position?
Question 6:
Suppose a1 , …, an are in m .
Is span {a1 , …, an } = m ?
Method:
This question is equivalent to Question 5 with A = [ a1 an ] .
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AMS 10 Mathematical Methods for Engineers I
So we look at the pivot positions of matrix A.
(Theorem 4 of Chapter 1)
Question 7:
Suppose a1 , …, an are in m .
Is every b in m a linear combination of a1 , …, an ?
Method:
This question is equivalent to Question 5 with A = [ a1 an ] .
So we look at the pivot positions of matrix A.
(Theorem 4 of Chapter 1)
Question 8:
Does A x = 0 have a non-trivial solution?
Method:
To answer this question, we look at the number of free variables of A x = 0 .
(Theorem 2 of Chapter 1)
Question 8B:
Is A x = 0 always consistent?
Answer: Yes. x = 0 is always a solution, which is called the trivial solution.
Question 8C:
Suppose matrix A is 11 15.
Does A x = 0 have a non-trivial solution?
Method:
Examine the number of free variables of A x = 0 .
Question 9:
Is {a1 , …, an } in m linearly dependent?
Method:
This question is equivalent to Question 8 with A = [ a1 an ] .
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AMS 10 Mathematical Methods for Engineers I
So we look at the number of free variables of A x = 0 .
(Theorem 2 of Chapter 1)
Questions 10-12 are the 3 special cases where we can determine linear dependence without
carrying out row reduction.
Question 10:
Is u1 , u2 , 0 linearly dependent?
{
}
(Theorem 9 of Chapter 1)
Question 11:
Is {u, v } linearly dependent?
Method:
To answer this question, we look at if one vector is a multiple of the other.
(Theorem 7 of Chapter 1)
Question 12:
Suppose a1 , …, an are in m .
Suppose n > m.
Is {a1 , …, an } linearly dependent?
(Theorem 8 of Chapter 1)
Question 13:
Suppose matrix A is m n.
Suppose n > m.
Does A x = 0 have a non-trivial solution?
Method:
This question is equivalent to Question 12 with {a1 , …, an } = columns of A.
(Theorem 8 of Chapter 1)
Alternatively, we can
answer this question directly by examining the number of free
variables of A x = 0 (which is the proof of Theorem 8, Chapter 1).
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AMS 10 Mathematical Methods for Engineers I
Linear systems, vectors, matrices (continued)
Row-vector rule for computing A x
Row-column rule for computing AB
The transpose of a matrix:
( A B )T = ?
(Theorem 3 of Chapter 2)
The inverse of a matrix:
Definition of inverse:
Suppose matrix A is n n.
If there exists C such that C A = I and A C = I, then we say A is invertible.
If A and B are both invertible, then AB is invertible and
( A B )1 = ?
(Theorem 6 of Chapter 2)
The algorithm for finding A-1 (and determining if A is invertible)
(Theorem 7 of Chapter 2)
Question 14:
Suppose AB is well defined. Does that imply BA is well defined?
Suppose AB = 0. Does that imply BA = 0?
Suppose AB = AC. Does that imply A = 0 or B = C?
Suppose matrix A is n n. Is A5 well defined?
Suppose A is 3 5 and B is 4 3. Is AB well defined? Is BA well defined? Is A7 well
defined? Is (AB)7 well defined?
Question 15:
Suppose A and B are both n n and are both invertible.
Is (A-1B-1) invertible? If so, what is (A-1B-1) -1 ?
Is (B A-1) invertible? If so, what is (B A-1)-1 ?
Theorem 8 of Chapter 2 (The invertible matrix theorem)
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AMS 10 Mathematical Methods for Engineers I
Question 16:
Suppose A = [ a1 an ] is an n n matrix and is invertible.
Does matrix A have n pivot positions?
Does A x = 0 have a non-trivial solution?
Is {a1 , …, an } linearly independent?
Is A x = b consistent for every b in n ?
span {a1 , …, an } = n ?
(Theorem 8 of Chapter 2)
Question 17:
0 1 2
A = 1 1 3 4 3 6 Let
How do we determine if matrix A is invertible (without carrying out the full algorithm of
finding the invers)?
Row reduction to an echelon form and check the number of pivot positions.
Method:
Question 18:
0 1 2
A = 1 1 3 4 3 6 Let
How do we find the inverse of matrix A?
Question 18B:
0 1 2
A = 1 1 3 4 3 6 Let
Suppose we found an answer for A-1 and we want to check our answer.
9 / 4 2 1 / 4 A = 3 / 2 2 1 / 2 5 / 4 1 1 / 4 1
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AMS 10 Mathematical Methods for Engineers I
Do we need to check both A 1 A = I and A A 1 = I ?
Question 19:
Let A be an n n matrix. Suppose there is a matrix C such that C A = I.
Is matrix A invertible?
If so, what is A 1 ?
(Theorem 8 of Chapter 2)
Question 19B:
Let A be an n n matrix. Suppose there is a matrix D such that A D = I.
Is matrix A invertible?
If so, what is A 1 ?
(Theorem 8 of Chapter 2)
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