MA 135 Applied Discrete Math Study Notes

1.1 Vectors

Definition of a Vector: A vector is a mathematical object that has both a magnitude and a direction.
- Example: $v = (1, 3)$
Vector Notation:
- In vector notation, $W=(-3,4) * 4$ represents a scaled vector.
- The formula in the context of vectors is referred to as Formula on 19.2.
Notation:
- Initial point of a vector $a=(-3,1)$
- Terminal point of a vector $a=(-1,4)$
- Standard position vector $v = (b1-a1, b2-a2)$
- A standard position vector has its origin as the initial point.
n-tuple: An ordered list of n numbers represented as $(a1, a2,…, a_n)$ .
- Example in real numbers: $v = (1, 3, -2)$ is a vector in $ ext{IR}^3$
Zero Vector: A vector where all components are zero, e.g., $(0,0,0,…,0)$
Applications of Vectors:
- Vectors are utilized to store data.
- Vectors are also used to create geometric shapes.

1.2 Finding Standard Position Vector

Finding the Standard Position Vector:
- Given an initial point $(a1, a2)$ and a terminal point $(b1, b2)$ , the standard position vector is:
 $V = (b1-a1, b2-a2)$
- This formula can be extended to any dimension where
 $V = (b1-a1, b2-a2, …, bn-an)$
Magnitude of a Vector: The length of vector $v$ , denoted by $ ext{||v||}$, is defined as:
$||v|| = ext{sqrt}(a1^2 + a2^2 + … + a_n^2)$
- Example: Finding the magnitude of vector with initial point $(1,5)$ and terminal point $(2,-4)$ .
- Standard position vector: $V = (2-1, -4-5) = (1, -9)$
- Magnitude:
 $||V|| = ext{sqrt}(1^2 + (-9)^2) = ext{sqrt}(1 + 81) = ext{sqrt}(82)$
Direction of a Vector: Described by the angle from the positive x-axis to the vector, measured in a counterclockwise direction.
- Example: For the vector $V = (2,2)$ , the angle is $45^ ext{o}$ .

1.3 Working with Directions

Finding Direction:
- Example: To find the direction of the vector from $(6, 1)$ to $(-2, -5)$ , standard position is:
  $V = (-2-6, -5-1) = (-8, -6)$
- The angle calculated as:
  $heta = 360^ ext{o} - 45^ ext{o} = 315^ ext{o}$
Specific Vector Calculations:
- Example: Find the direction and magnitude of a vector starting from $(-2,5)$ and ending at $(-3,-1)$ :
- Standard position vector $V = (-3 - (-2), -1 - 5) = (-1, -6)$
- Magnitude:
  $||V|| = ext{sqrt}((-1)^2 + (-6)^2) = ext{sqrt}(1 + 36) = ext{sqrt}(37)$
- Direction $ heta$:
  $heta = ext{tan}^{-1} rac{-6}{-1} = 80.5^ ext{o}, ext{ then } heta = 180 + 80.5 = 260.5^ ext{o}$
Examples of Other Vectors:
- Example with vector $V = (-3,-5)$ results in:
  $heta = 180 + 59.03 = 239.03^ ext{o}$
- For vector $w = (-4,5)$ ,
  $heta = 180 - 51.3 = 128.7^ ext{o}$

1.4 Vector Review

Vector Representation: In vector notation, $v = [7]$ signifies a vertical vector with length $ ext{||v||} = ext{sqrt}(a1^2 + a2^2 + … + a_n^2)$
Direction: The direction of vector $v$ is determined as the angle from the positive x-axis measured counterclockwise.
- Example: $w=(-3,5)$ which needs to be analyzed for direction.

1.5 Vector Operations

Scalar Multiplication:
- If $k$ is a constant and $v$ is a vector in $ ext{IR}$, then:
 $k imes v = k (a1, a2, a3) = (k a1, k a2, k a3, ext{…, } k a_n)$
- If $k > 0$ , then the result is a stretch of $v$ , if $0<k<1$ , it's a shrink, and if $k < 0$ , the resultant vector is scaled in the opposite direction of $v$ .
Adding and Subtracting Vectors: For vectors in the same dimension, use:
$V + W = (a1 + b1, a2 + b2, …, an + bn)$
$V - W = (a1 - b1, a2 - b2, …, an - bn)$
Geometrical Representation:
- Refer to graphical illustrations for visualization.

1.6 Unit Vector (Normalizing)

Definition: A unit vector has a length equal to 1.
- Given vector $v$ , to find its unit vector:
  $u = rac{v}{||v||}$
- Example for $v = (8, -2)$ :
  - Length calculation leads to unit vector $u$ .

1.7 Dot Product of Vectors

Definition: For vectors $v$ and $w$ , the dot product is defined as:
$v \bullet w = ext{sum}(ai bi) = (a1 b1 + a2 b2 + … + an bn)$
- Example computation.
Properties of Dot Products:
1. Commutative: $u \bullet v = v \bullet u$
2. Distributive: $u \bullet (v + w) = u \bullet v + u \bullet w$
3. Associative: $(cu)v = c(u \bullet v)$
Geometric Interpretation:
$u \bullet v = ||u|| imes ||v|| imes ext{cos}( heta)$ where $ heta$ is the angle between vectors.
Example Calculations: Using specific numbers for vectors to show detailed numerical operations.

1.8 Orthogonal Vectors

Definition: Two vectors are orthogonal if the angle between them is $90^ ext{o}$ .
- Condition: $u \bullet v = 0$ .
Scalar Projection: The scalar projection of vector $V$ onto vector $W$ , denoted as:
$ext{Comp}_W V = rac{V \bullet W}{||W||}$
Further Explanation: Deriving the projection formula based on vector relations.

1.9 Vector Projection

Definition: The vector projection of $u$ onto $v$ is achieved:
$ext{Proj}_{v} u = rac{u \bullet v}{||v||^2} v$
Examples: Working through numeric samples.

2. Matrices

Definition of a Matrix: A matrix is an array of numbers represented as:
A = egin{bmatrix} a{11} & a{12} & … & a{1n} \ a{21} & a{22} & … & a{2n} \ … & … & … & … \ a{m1} & a{m2} & … & a_{mn} \ ext{where } m ext{ is the number of rows and } n ext{ is the number of columns} \ ext{Example: } A = egin{bmatrix} -3 & 5 \ 3 & 2 \ ext{This indicates a } 2 imes 2 ext{ matrix.}
Matrix Types:
- Zero Matrix: All elements are zero.
- Identity Matrix: Forms diagonal of ones.
Operations on Matrices:
- Scalar Multiplication:
 k imes A = k egin{bmatrix} a{11} & a{12} \ a{21} & a{22} \ ext{will lead to each element being scaled by } k. $</li><li>Addition: Matrices can be added if they have the same dimensions.</li></ul></li><li>Properties of Matrix Addition and Multiplication:<ol><li>Commutative Property</li><li>Associative Property</li></ol></li></ul><h4 id="a705b996-3bcf-48a1-b32c-96df986a359b" data-toc-id="a705b996-3bcf-48a1-b32c-96df986a359b" collapsed="false" seolevelmigrated="true">3. Systems of Linear Equations</h4><ul><li>Definition: A linear equation takes the form: $ a1x1 + a2x2 + … + anxn = b $<ul><li>Example:$ x + y = 1 $is a linear equation.</li></ul></li><li>Matrix Representation: A system can be represented as: $ AX = B $where$ A $is the coefficient matrix and$ B $is the constant matrix.</li><li>Solving Systems: Use techniques such as EROS (Elementary Row Operations) to transform matrices.</li><li>Example Problem: Show steps taken to solve a system of equations.</li></ul><h4 id="bdb53bc6-a8c5-4301-a8c9-a257f7eb85f3" data-toc-id="bdb53bc6-a8c5-4301-a8c9-a257f7eb85f3" collapsed="false" seolevelmigrated="true">4. Integer Properties</h4><ul><li>Types of Numbers:<ul><li>Natural Numbers: positive whole integers.</li><li>Integers: positive, negative whole numbers, and zero.</li><li>Rational Numbers: can be expressed as the quotient of two integers.</li><li>Real Numbers: includes both rational and irrational numbers.</li></ul></li><li>Divisibility: An integer$ k $divides integer$ m $if: $ m = nk ext{ for some integer } n. $<ul><li>Example:$ 2 $divides$ 6 $.</li></ul></li><li>Divisibility Definitions and Notations.<ul><li>Linear combinations, division algorithms, and existence of unique integers pertaining to modular arithmetic.</li></ul></li></ul><h4 id="e370cd75-4b9a-4cb1-adf7-f0555934abb2" data-toc-id="e370cd75-4b9a-4cb1-adf7-f0555934abb2" collapsed="false" seolevelmigrated="true">5. Modular Arithmetic</h4><ul><li>Definition: Given an integer$ m > 1 $,$ x ext{ mod } m $represents the remainder when$ x $is divided by$ m $.</li><li>Properties: Different theorems based on mod operations.</li><li>Example of Modular Operations: Show examples of addition and multiplication in modular systems.</li><li>Theorem Summary: Briefly detail important theorems regarding modular arithmetic.</li></ul><h4 id="717f08c5-33d0-494a-9a2c-88092b9beaf6" data-toc-id="717f08c5-33d0-494a-9a2c-88092b9beaf6" collapsed="false" seolevelmigrated="true">6. Prime Factorization</h4><ul><li>Definition of a Prime Number: An integer greater than 1, divisible only by itself and 1.</li><li>Fundamental Theorem of Arithmetic: Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers.</li><li>Common Applications: Finding the greatest common divisor (gcd) and least common multiple (lcm) through prime factorization.</li></ul><ol><li>Standard Position Vector: $ V = (b1 - a1, b2 - a2) $ For any dimension: $ V = (b1 - a1, b2 - a2, …, bn - an) $</li><li>Magnitude of a Vector: $ ||v|| = \sqrt{a1^2 + a2^2 + … + a_n^2} $</li><li>Vector Direction: $ theta=\text{angle from the positive x-axis measured counterclockwise} $</li><li>Scalar Multiplication: $ k \times v = k (a1, a2, a3) = (k a1, k a2, k a3, …, k a_n) $</li><li>Adding and Subtracting Vectors: $ V + W = (a1 + b1, a2 + b2, …, an + bn) $ $ V - W = (a1 - b1, a2 - b2, …, an - bn) $</li><li>Unit Vector: $ u = \frac{v}{||v||} $</li><li>Dot Product: $ v • w = \text{sum}(ai bi) = (a1 b1 + a2 b2 + … + an bn) $</li><li>Geometric Interpretation of Dot Product: $ u • v = ||u|| \times ||v|| \times \cos(\theta) $</li><li>Scalar Projection: $ \text{Comp}_W V = \frac{V • W}{||W||} $</li><li>Vector Projection: $ \text{Proj}_{v} u = \frac{u • v}{||v||^2} v $</li><li>Linear Equation: $ a1 x1 + a2 x2 + … + an xn = b $</li><li>Matrix Representation of a System: $ AX = B $</li><li>Divisibility: $ m = nk \text{ for some integer } n $</li><li>Modular Arithmetic: $ x \text{ mod } m = \text{remainder when } x \text{ is divided by } m $</li></ol><ul><li>Operations on Matrices:<ul><li>Scalar Multiplication:$ k imes A = k \begin{bmatrix} a{11} & a{12} \ a{21} & a{22} \end{bmatrix} \text{ will lead to each element being scaled by } k.
- Addition: Matrices can be added if they have the same dimensions.
- Example of Matrix Addition:
 Given two matrices:
 A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \ B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix},
 Then,
 A + B = \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}.
Matrix Representation:
A system can be represented as:

AX = B $where$ A $is the coefficient matrix and$ B $is the constant matrix.</li><li>Example Problem: Show steps taken to solve a system of equations. The specific calculations for this will depend on the provided systems of equations, but general steps include:<ol><li>Set up the augmented matrix </li><li>Use techniques such as EROS (Elementary Row Operations) to transform matrices.</li></ol></li><li>Properties of Matrix Addition and Multiplication:<ol><li>Commutative Property</li><li>Associative Property</li><li>Distributive Property </li></ol></li></ul><h5 id="ae91b10a-e658-4ece-a4a6-1ddca2ee889e" data-toc-id="ae91b10a-e658-4ece-a4a6-1ddca2ee889e" collapsed="false" seolevelmigrated="true">Matrix Multiplication</h5><ul><li>Definition: Matrix multiplication involves the multiplication of two matrices to produce a third matrix. The number of columns in the first matrix must equal the number of rows in the second matrix.</li><li>Matrix Representation: If we have two matrices, A and B, and we want to multiply them:$ A = \begin{bmatrix} a{11} & a{12} & … & a{1n} \ a{21} & a{22} & … & a{2n} \ … & … & … & … \ a{m1} & a{m2} & … & a_{mn} \end{bmatrix} $$ B = \begin{bmatrix} b{11} & b{12} & … & b{1p} \ b{21} & b{22} & … & b{2p} \ … & … & … & … \ b{n1} & b{n2} & … & b_{np} \end{bmatrix} $</li><li>Resulting Matrix: The product C of matrices A and B will be:$ C = AB = \begin{bmatrix} c{11} & c{12} & … & c{1p} \ c{21} & c{22} & … & c{2p} \ … & … & … & … \ c{m1} & c{m2} & … & c_{mp} \end{bmatrix} $where each element$ c_{ij} $is calculated as:$ c{ij} = \sum{k=1}^{n} a{ik} b{kj} $</li><li>Example Calculation:Consider:$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \, B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} $</li><li>To find the product AB, compute:$ C = AB = \begin{bmatrix} (15 + 27) & (16 + 28) \ (35 + 47) & (36 + 48) \end{bmatrix} $Simplifying this gives:$ C = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix} $</li><li>Properties of Matrix Multiplication:<ol><li>Not Commutative: Generally,$ AB \neq BA $.</li><li>Associative:$ (AB)C = A(BC) $.</li><li>Distributive:$ A(B + C) = AB + AC$$.

Summary

Ensure the dimensions are suitable before multiplication and follow the specific rule for calculating the resulting matrix based on summing products of corresponding elements.