MA 135 Applied Discrete Math Study Notes

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:
    $V = (b<em>1-a</em>1, b<em>2-a</em>2)$

  • This formula can be extended to any dimension where
    $V = (b<em>1-a</em>1, b<em>2-a</em>2, …, b<em>n-a</em>n)$

• Magnitude of a Vector: The length of vector $v$, denoted by $ ext{||v||}$, is defined as:
  $||v|| = ext{sqrt}(a<em>1^2 + a</em>2^2 + … + a_n^2)$

  • 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:
    $k imes v = k (a<em>1, a</em>2, a<em>3) = (k a</em>1, k a<em>2, k a</em>3, ext{…, } k a_n)$

  • 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:
  $V + W = (a<em>1 + b</em>1, a<em>2 + b</em>2, …, a<em>n + b</em>n)$
  $V - W = (a<em>1 - b</em>1, a<em>2 - b</em>2, …, a<em>n - b</em>n)$

• 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 ullet w = ext{sum}(ai bi) = (a1 b1 + a2 b2 + … + an bn)

  • Example computation.

• Properties of Dot Products:

  1. Commutative: $u ullet v = v ullet u$

  2. Distributive: $u ullet (v + w) = u ullet v + u ullet w$

  3. Associative: $(cu)v = c(u ullet v)$

• Geometric Interpretation:
  u ullet 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: $u ullet v = 0$.

• Scalar Projection: The scalar projection of vector $V$ onto vector $W$, denoted as:
  ext{Comp}_W V = rac{V ullet 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 ullet 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. $</p></li><li><p><strong>Addition</strong>: Matrices can be added if they have the same dimensions.</p></li></ul></li><li><p><strong>Properties of Matrix Addition and Multiplication</strong>:</p><ol><li><p>Commutative Property</p></li><li><p>Associative Property</p></li></ol></li></ul><h4 id="a705b996-3bcf-48a1-b32c-96df986a359b" data-toc-id="a705b996-3bcf-48a1-b32c-96df986a359b" collapsed="false" seolevelmigrated="true">3. Systems of Linear Equations</h4><ul><li><p><strong>Definition</strong>: A linear equation takes the form:<br>$ a1x1 + a2x2 + … + anxn = b

    • Example: $x + y = 1$ is a linear equation.

  • Matrix Representation: A system can be represented as:
    AX = B where $A$ is the coefficient matrix and $B$ is the constant matrix.

  • Solving Systems: Use techniques such as EROS (Elementary Row Operations) to transform matrices.

  • Example Problem: Show steps taken to solve a system of equations.

  4. Integer Properties

  • Types of Numbers:

    • Natural Numbers: positive whole integers.

    • Integers: positive, negative whole numbers, and zero.

    • Rational Numbers: can be expressed as the quotient of two integers.

    • Real Numbers: includes both rational and irrational numbers.

  • Divisibility: An integer $k$ divides integer $m$ if:
    m = nk ext{ for some integer } n.

    • Example: $2$ divides $6$.

  • Divisibility Definitions and Notations.

    • Linear combinations, division algorithms, and existence of unique integers pertaining to modular arithmetic.

  5. Modular Arithmetic

  • Definition: Given an integer $m > 1$, $x ext{ mod } m$ represents the remainder when $x$ is divided by $m$.

  • Properties: Different theorems based on mod operations.

  • Example of Modular Operations: Show examples of addition and multiplication in modular systems.

  • Theorem Summary: Briefly detail important theorems regarding modular arithmetic.

  6. Prime Factorization

  • Definition of a Prime Number: An integer greater than 1, divisible only by itself and 1.

  • Fundamental Theorem of Arithmetic: Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers.

  • Common Applications: Finding the greatest common divisor (gcd) and least common multiple (lcm) through prime factorization.

  1. Standard Position Vector:
     V = (b1 - a1, b2 - a2) $<br>For any dimension: <br>$ V = (b1 - a1, b2 - a2, …, bn - an) $</p></li><li><p><strong>Magnitude of a Vector</strong>: <br>$ ||v|| = \sqrt{a1^2 + a2^2 + … + a_n^2} $</p></li><li><p><strong>Vector Direction</strong>: <br>$theta=\text{angle from the positive x-axis measured counterclockwise}$</p></li><li><p><strong>Scalar Multiplication</strong>: <br>$ k \times v = k (a1, a2, a3) = (k a1, k a2, k a3, …, k a_n) $</p></li><li><p><strong>Adding and Subtracting Vectors</strong>: <br>$ V + W = (a1 + b1, a2 + b2, …, an + bn) $<br>$ V - W = (a1 - b1, a2 - b2, …, an - bn) $</p></li><li><p><strong>Unit Vector</strong>: <br>$ u = \frac{v}{||v||} $</p></li><li><p><strong>Dot Product</strong>: <br>$ v • w = \text{sum}(ai bi) = (a1 b1 + a2 b2 + … + an bn) $</p></li><li><p><strong>Geometric Interpretation of Dot Product</strong>: <br>$ u • v = ||u|| \times ||v|| \times \cos(\theta) $</p></li><li><p><strong>Scalar Projection</strong>: <br>$ \text{Comp}_W V = \frac{V • W}{||W||} $</p></li><li><p><strong>Vector Projection</strong>: <br>$ \text{Proj}_{v} u = \frac{u • v}{||v||^2} v $</p></li><li><p><strong>Linear Equation</strong>: <br>$ a1 x1 + a2 x2 + … + an xn = b $</p></li><li><p><strong>Matrix Representation of a System</strong>: <br>$ AX = B $</p></li><li><p><strong>Divisibility</strong>: <br>$ m = nk \text{ for some integer } n $</p></li><li><p><strong>Modular Arithmetic</strong>: <br>$ x \text{ mod } m = \text{remainder when } x \text{ is divided by } m $</p></li></ol><p></p><ul><li><p><strong>Operations on Matrices</strong>:</p><ul><li><p><strong>Scalar Multiplication</strong>:</p><p>$ 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. $</p></li><li><p><strong>Addition</strong>: Matrices can be added if they have the same dimensions.</p></li><li><p><strong>Example of Matrix Addition</strong>:</p><p>Given two matrices: <br>$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \ B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}, $<br> Then,<br>$ A + B = \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}. $</p></li></ul></li><li><p><strong>Matrix Representation</strong>:</p><p>A system can be represented as:<br><br>$ AX = B where $A$ is the coefficient matrix and $B$ is the constant matrix.

  2. 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:

     1. Set up the augmented matrix

     2. Use techniques such as EROS (Elementary Row Operations) to transform matrices.

  3. Properties of Matrix Addition and Multiplication:

     1. Commutative Property

     2. Associative Property

     3. Distributive Property

Matrix Multiplication

• 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.

• 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} $</p><p>$ B = \begin{bmatrix} b{11} & b{12} & … & b{1p} \ b{21} & b{22} & … & b{2p} \ … & … & … & … \ b{n1} & b{n2} & … & b_{np} \end{bmatrix} $</p></li><li><p><strong>Resulting Matrix</strong>: The product C of matrices A and B will be:</p><p>$ 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} $</p></li><li><p><strong>Example Calculation</strong>:</p><p>Consider:</p><p>$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \, B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} $</p></li><li><p>To find the product AB, compute:</p><p>$ C = AB = \begin{bmatrix} (15 + 27) & (16 + 28) \ (35 + 47) & (36 + 48) \end{bmatrix} $</p><p>Simplifying this gives:</p><p>$ C = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix} $$

• Properties of Matrix Multiplication:

  1. Not Commutative: Generally, $AB \neq BA$.

  2. Associative: $(AB)C = A(BC)$.

  3. 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.