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Introduction to Vectors and Operations

What is a Vector?

  • A vector is a mathematical object represented as an ordered set of numbers enclosed by brackets.

    • Example: ( \mathbf{a} = \begin{bmatrix} 2 \ 1 \end{bmatrix} )

    Vectors are denoted in various notations but represent the same concept. For example:

    • ( a=(2,1)\mathbf{a} = (2, 1) )
    • ( a=2,1\mathbf{a} = { 2, 1 } )
    • ( a=[2 1]\mathbf{a} = \left[ 2 \ 1 \right] )

  • Dimension of Vectors:

    • A vector can be defined in 1-dimensional or 2-dimensional space.
    • 1D Vector: ( R\mathbb{R} )
    • 2D Vector: ( R2\mathbb{R}^2 )

  • A vector by itself does not convey much information. It only represents a set of numbers enclosed in brackets, and it becomes meaningful when additional context (like direction or position) is given.

Graphical Representation of Vectors

  • A vector can be represented on a graph by indicating the direction and magnitude.
    • Example: A vector ( a=[3 4]\mathbf{a} = \begin{bmatrix} 3 \ 4 \end{bmatrix} ) on a 2D plane.

Vector Operations

Vector Addition

  • For any two vectors ( A\mathbf{A} ) and ( B\mathbf{B} ) in an ( nn )-dimensional space:

    • The addition is performed component-wise.
    • Formula: ( A+B=[A1+B1 A2+B2 ⋮ An+Bn]\mathbf{A} + \mathbf{B} = \begin{bmatrix} A_1 + B_1 \ A_2 + B_2 \ \vdots \ A_n + B_n \end{bmatrix} )

  • Example:

    • ( A=[2 3],B=[3 2]\mathbf{A} = \begin{bmatrix} 2 \ 3 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} 3 \ 2 \end{bmatrix} )
    • ( A+B=[2+3 3+2]=[5 5]\mathbf{A} + \mathbf{B} = \begin{bmatrix} 2 + 3 \ 3 + 2 \end{bmatrix} = \begin{bmatrix} 5 \ 5 \end{bmatrix} )

Vector Subtraction

  • Similar to addition, vector subtraction is performed component-wise.

    • Formula: ( A−B=[A1−B1 A2−B2 ⋮ An−Bn]\mathbf{A} - \mathbf{B} = \begin{bmatrix} A_1 - B_1 \ A_2 - B_2 \ \vdots \ A_n - B_n \end{bmatrix} )

  • Example:

    • ( A=[3 4],B=[2 1]\mathbf{A} = \begin{bmatrix} 3 \ 4 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} 2 \ 1 \end{bmatrix} )
    • ( A−B=[3−2 4−1]=[1 3]\mathbf{A} - \mathbf{B} = \begin{bmatrix} 3 - 2 \ 4 - 1 \end{bmatrix} = \begin{bmatrix} 1 \ 3 \end{bmatrix} )

The Transpose Operation

  • The transpose of a vector involves flipping it, turning rows into columns and vice versa.

    • Formula: If ( x=[A1 A2 … An]),then(xT=[A1A2…An]\mathbf{x} = \begin{bmatrix} A_1 \ A_2 \ \dots \ A_n \end{bmatrix} ), then ( \mathbf{x}^T = \begin{bmatrix} A_1 & A_2 & \dots & A_n \end{bmatrix} )

  • Example:

    • ( x=[1 3]\mathbf{x} = \begin{bmatrix} 1 \ 3 \end{bmatrix} )
    • Transpose: ( xT=[13]\mathbf{x}^T = \begin{bmatrix} 1 & 3 \end{bmatrix} )

Key Concepts

  • Vector notation: Used to represent magnitude and direction.
  • Operations: Addition, subtraction, and transposition can be performed on vectors.
  • Meaning: A vector itself doesn’t convey much until you provide context or perform operations on it.