LectureNotes2-24

Chapter 2: Vectors and Matrices

  • Introduction to MATLAB

    • MATLAB stands for Matrix Laboratory.

    • Designed to work effectively with vectors and matrices.

Vectors and Matrices

  • Definition:

    • Vectors and matrices store sets of values of the same type.

    • Example representations:

      • Scalar: 32 (1x1 matrix)

      • Row vector: [5 88 3 11] (1x4 matrix)

      • Column vector: [3.63; 7.28; 4.59] (6x1 matrix)

      • Matrix:

        [  42  19  23  13  
  24  82  77  0]

        (3x4 matrix)

  • Matrix Representation:

    • Values must align in rows.

Creating Row Vectors

  • Vector Assignment Examples:

    • v = [1 2 3 4]

    • v = [1, 2, 3, 4]

  • Colon Operator:

    • Generates a vector.

    • Example:

      • vec = 1:5

      • Result: [1 2 3 4 5]

  • Step Value Usage:

    • Specify a step size:

      • v = 1:2:9

        • Result: [1 3 5 7 9]

Creating Vectors with Decimal Numbers

  • Decimal Example:

    • v = 1.1:2.3:8

  • Using linspace:

    • Syntax: v = linspace(x, y, n)

      • Creates a linearly spaced vector with 'n' values from 'x' to 'y'.

      • Example: v = linspace(3,15,5) results in [3 6 9 12 15].

Creating Column Vectors

  • Column Vector Definition:

    • Defined by placing values in brackets and separating with semicolons.

    • Example: c = [1; 2; 3; 4] results in:

    1  
2  
3  
4

Modifying Vectors and Elements

  • Indexing:

    • Elements are indexed starting at 1.

    • Example: vec = [1 3 5 7 9]

      • Access: vec(5) results in 9.

  • Subsetting Vectors:

    • Using colon operator: b = vec(4:6) results in [7 9 3].

  • Index Vector Example:

    • c = vec([1 10 5]) yields values from specified locations.

Vector Assignment and Extending

  • Changing Values:

    • Assigning new values: b(2) = 11 changes the second value.

  • Extending Vectors:

    • rv(4) = 2 extends the vector.

  • Handling Gaps:

    • Gaps will be filled with zeros when extending.

Working with Matrices

  • Matrix Creation Example:

    • mat = [4 3 1; 2 5 6] defines a 2x3 matrix.

  • Random Matrices:

    • Use rand for random numbers: A = rand(2).

  • Creating Identity Matrices:

    • eye(3) creates a 3x3 identity matrix.

Modifying Matrix Elements

  • Element Modification:

    • mat(2,3)=6 changes an element.

  • Row/Column Modification:

    • Change entire row: mat(2,:) = 5:7.

  • Adding Rows and Columns:

    • mat(:,4) = [9 2]' adds a new column.

Dimensions and Size Properties

  • Length and Size:

    • length(vec) gives the number of elements in a vector.

    • size(mat) returns number of rows and columns in a matrix.

    • Use [r,c] = size(mat) to store results.

Reshaping Matrices

  • Using reshape:

    • Changes dimensions without altering data.

    • Example: reshape(mat,2,6) results in different dimensional arrangements.

Empty Vectors and Concatenation

  • Creating Empty Vectors:

    • evec = [] defines an empty vector.

  • Adding Values to Empty Vectors:

    • evec = [evec 4] adds values to 'evec'.

  • Deleting Elements:

    • Use indexing to remove elements: vec(3) = [].

  • Removing Subsets:

    • E.g., vec(2:4) = [] removes part of the vector.

Three-Dimensional Matrices

  • Creating Layers:

    • Use reshape for layers: layerone = reshape(1:15,3,5).

Functions with Vectors and Matrices

  • Basic Functions:

    • sum, min, max work with both vectors and matrices.

  • Finding Elements:

    • Use find() to locate indices meeting criteria, e.g., find(vec > 5).

Logical Operations

  • Using Logical Indexing:

    • Create boolean vectors to filter data.

  • Functions:

    • Use any, all, isequal for logical conditions within vectors/matrices.

Summary of Operations

  • Scalar and Array Operations:

    • Operations can be performed element-wise.

  • Matrix Multiplication:

    • Requires matching inner dimensions.

Miscellaneous

  • Meshgrid Function:

    • Helps create coordinate grids for functions.

  • Trace Function:

    • Returns sum of the diagonal elements of a square matrix.

Extensions and Homework Tips

  • Experiment with vector and matrix operations, including dot, cross, and reshaping.

  • Implement logical indexing for data analysis.

  • Practice creating and manipulating random and structured matrices.