Lecture 3

Introduction to Vectors and Matrices

  • Focus on fundamental aspects of MATLAB for working with vectors and matrices, important for data analysis and engineering problems.

Overview of MATLAB

  • MATLAB stands for "Matrix Laboratory."

  • Designed to work efficiently with vectors and matrices.

Mathematical Concepts

Scalars, Vectors, and Matrices

  • Scalar: A matrix of size 1x1, containing a single element.

  • Row Vector: Defined as a matrix with a single row and multiple columns (e.g., a 1x4 matrix).

    • Example:

    • Row vector representation:

      • [5, 88, 3, 11] is a 1x4 matrix.

  • Column Vector: A matrix with multiple rows and a single column (e.g., a 6x1 matrix).

    • Example:

    • Column vector representation:

      • egin{bmatrix} a1 \ a2 \ a3 \ a4 \ a5 \ a6 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{bmatrix} where each element corresponds to a different data point.

  • Matrix: A collection of numbers arranged in a rectangular format; defined by its number of rows and columns.

    • Example: A 3x4 matrix has 3 rows and 4 columns.

MATLAB Arrays and Vectors Creation

Creating Vectors

  • A vector can be defined in MATLAB:

    • Use square brackets, e.g., V = [1, 2, 3, 4] or V = [1 2 3 4] to create a row vector.

    • To create a column vector, use semicolons between elements:

    • Example: C = [1; 2; 3; 4] creates a column vector.

Using the Colon Operator

  • A powerful operator in MATLAB for creating sequences:

    • Example: 1:5 creates an array of integers from 1 to 5.

    • Custom increments can be specified:

    • Example: 1:2:9 generates [1, 3, 5, 7, 9] (from 1 to 9, incrementing by 2).

Using linspace Function

  • Generates a linearly spaced vector.

  • Syntax: linspace(X, Y, N) creates a vector of N points between X and Y.

    • Example: linspace(3, 15, 5) produces the vector [3, 6, 9, 12, 15].

Using logspace Function

  • Creates logarithmically spaced vectors (not discussed in detail).

    • Example: logspace(1, 3, 5) gives [10^1, 10^2, 10^3, 10^4, 10^5].

Basic Vector Operations in MATLAB

Concatenating Vectors

  • Combine vectors:

    • Example: V = [V1, V2] where V1 and V2 are vectors.

Indexing

  • Access specific elements using indexes:

    • vec(i) retrieves the ith element of a vector.

    • Accessing ranges: vec(a:b) provides elements from index a to b.

  • Example of non-consecutive indexing:

    • vec([1, 5, 10]) retrieves the first, fifth, and tenth elements.

Modifying Vector Elements

  • Changing elements:

    • V(i) = value changes the value at index i.

  • Expanding vectors by assigning a value to a new index:

    • Example: V(6) = 13 adds an element (if none exists, MATLAB fills it with 0 automatically).

Row and Column Vectors

  • Row vectors can be transposed to become column vectors:

    • Use the prime operator, e.g., R' (where R is a row vector).

  • Creating column vectors using semicolon operators is essential in MATLAB.

Creating Matrix Variables

Defining Matrices

  • Matrices can be created using similar notation as vectors:

    • Example: M = [4, 3, 1; 2, 5, 6] for a 2x3 matrix.

Accessing Matrix Elements

  • Access specific values using two indices: M(i,j) where i is the row and j is the column.

  • Example: M(2,2) retrieves the value in the second row and second column.

Reshaping Matrices

  • Use the reshape function to change the dimensions of a matrix.

    • Example: reshape(M, 2, 6) reshapes it into a 2x6 matrix, filled column-wise.

Special Matrices in MATLAB

Creation of Special Matrices

  • Zero Matrix: Use the function zeros(m,n) to create an m-by-n matrix of zeros.

  • Ones Matrix: Use the function ones(m,n) to create an m-by-n matrix of ones.

  • Identity Matrix: Use eye(n) to create an n-by-n identity matrix.

Important Functions for Vectors and Matrices

Length and Size Functions

  • The function length(vector) returns the number of elements in a vector.

  • The function size(matrix) returns the number of rows and columns.

  • The function numel(matrix) returns the total number of elements in the matrix.

Modifying Matrices

  • Change entire rows/columns:

    • Example: M(row, :) = [new_values] replaces all elements in a specified row.

  • Expand matrices by defining a new row or new column:

    • Specify values for a new row or column corresponding to existing dimensions.

Conclusion and Next Steps

  • Encouragement to practice and explore further on functions in MATLAB.

  • Overview of upcoming assignments and study resources.

  • Emphasis on leveraging MATLAB capabilities for practical engineering analysis and data processing.

  • Reminder to finish homework and engage with upcoming sessions.