Notes 2

CHAPTER 2: VECTORS AND MATRICES

Overview of MATLAB

• MATLAB: abbreviation for "matrix laboratory".

• Designed to work superbly with vectors and matrices.

Vectors and Matrices

• Definition: Vectors and matrices are structures used to store sets of values that are of the same type.

• Examples of Vectors and Matrices:

  • Scalar (1x1 matrix): 5

  • Row vector (1x4 matrix): 1 2 3 4

  • Column vector (6x1 matrix):
    \begin{bmatrix} 3.63 \ 7.28 \ 4.59 \ 2 \ 42 \ 19 \end{bmatrix}

  • Matrix (3x4 matrix):
    \begin{bmatrix} 5 & 20 & 9.92 & 5.22 \ 98 & 33 & 21 & 24 \ 13 & 24 & 82 & 77 \end{bmatrix}

Creating Row Vectors

• Examples to create the same vector v:

  • Using square brackets:

  • v = [1 2 3 4]

  • v = [1, 2, 3, 4]

  • Both lines of code will produce:

  \text{v} = [1, 2, 3, 4]

Using the Colon Operator and Linspace Function

• Colon Operator: Generates a vector with integer steps:

  • vec = 1:5 yields:
    \text{vec} = [1, 2, 3, 4, 5]

• Specifying step value:

  - Syntax: first:step:last

  Example:

  • v = 1:2:9 produces:
    \text{v} = [1, 3, 5, 7, 9]

  • v = 9:-2:1 gives:
    \text{v} = [9, 7, 5, 3, 1]

• Decimal values:

  • Example: v = 1.1:2.3:8 produces:
    \text{v} = [1.1, 3.4, 5.7]

Linspace Function

• Definition: Creates a linearly spaced vector.

• Syntax: linspace(x, y, n) generates n values in the inclusive range from x to y:

  • Example: v = linspace(3, 15, 5

  • Output:
    \text{v} = [3, 6, 9, 12, 15]

• Conditions:

  • x, y can be integers or decimals, understand that n must be a positive integer.

Logspace Function

• Definition: Creates a logarithmically spaced vector.

• Example:

  • v = logspace(1, 5, 5) yields:
    \text{v} = [10, 100, 1000, 10000, 100000]

• Functionality: logspace(x1, x2, n) generates n points from $10^{x1}$ to $10^{x2}$.

Concatenating Vectors

• Definition: Putting two vectors together to create a new vector.

• Example:

  • Vectors v1 = [-2 0 2 4 6 8] and v2 = [10 3 6 9 12 15] can be concatenated:

  \text{newvec} = [v1 \, v2]

  • Result:
    \text{newvec} = [-2, 0, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15]

Referring to and Modifying Vector Elements

• Indexing: Elements in a vector are indexed sequentially; MATLAB uses 1-based indexing.

• Example of Vector Indexing:

  • For vector vec = [1, 3, 5, 7, 9, 3, 6, 9, 12, 15],

  • vec(5) returns:
    \text{ans} = 9

• Using the Colon Operator: To obtain a subset:

  • Example: b = vec(4:6) results in:
    \text{b} = [7, 9, 3]

Non-sequential Indices

• Example of using an index vector:

  • c = vec([1, 10, 5]) gives:
    \text{c} = [1, 15, 9]

Modifying Values in a Vector

• To change a value, the index can be specified:

  • Example: b(2) = 11 results in:
    \text{b} = [7, 11, 3]

• Vectors can be extended by referring to non-existing indices; gaps will be filled with zeros:

  • Example:

  • rv(4) = 2 followed by rv(6) = 13 results in:
    \text{rv} = [3, 55, 11, 2, 0, 13]

Creating Column Vectors

• To create a column vector directly:

  • Use square brackets, separating values with semicolons:
    c = [1; 2; 3; 4]

  • Output:
    \text{c} = \begin{bmatrix} 1 \ 2 \ 3 \ 4 \end{bmatrix}

• Transposing a row vector to a column vector is achieved using the apostrophe:

  • Example:

  • r = 1:3; c = r' gives:
    \text{c} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}

Creating Matrix Variables

• Matrices can be created explicitly:

  • Example:

  • mat = [4 3 1; 2 5 6] results in:
    \text{mat} = \begin{bmatrix} 4 & 3 & 1 \ 2 & 5 & 6 \end{bmatrix}

• Rule: The number of values in each row must be consistent.

Modifying Matrix Elements

• Create and modify a matrix element as follows:

  • Start matrix:

  • mat = [2:4; 3:5], results in:
    \text{mat} = \begin{bmatrix} 2 & 3 & 4 \ 3 & 4 & 5 \end{bmatrix}

  • Set an element:

  • mat(2, 3) = 7 changes the matrix to:
    \text{mat} = \begin{bmatrix} 2 & 3 & 4 \ 3 & 4 & 7 \end{bmatrix}

• Changing entire rows: Example: mat(2,:) = 5:7 to change the second row.

Adding Columns and Rows to Matrices

• To add a fourth column:

  • mat(:,4) = [9 2]' transforms mat into:
    \text{mat} = \begin{bmatrix} 2 & 3 & 4 & 9 \ 5 & 6 & 7 & 2 \end{bmatrix}

• To add a new third row:

  • mat(3,:)=[4 7 1 0] changes the matrix accordingly.

Dimensions of Arrays

• Length Function:

  • Returns the number of elements in a vector.

  • Example: For vec = -2:1, length(vec) yields:
    \text{ans} = 4

• Size Function: Returns the number of rows and columns in a matrix:

  • Example: For mat = [1:3; 5:7]', the size is:
    \text{size(mat)} = [3, 2]

• Syntax to retrieve dimensions into variables:

  • Example:

  • [r, c] = size(mat) then r = 3 and c = 2.

Numel Function

• Returns total number of elements in any array:

  • For a vector: numel(vec) gives the count of its entries:

Reshaping Matrices

• Reshape Function: Handles changing matrix dimensions.

• Example demonstrating a 3x4 reshaping into a 2x6:

  • Initial matrix mat = randi(100, 3, 4) yields a 3x4 matrix and can be reshaped:

  • reshape(mat, 2, 6) changes dimensions as specified.

Empty Vectors

• To create an empty vector, use evec = [] that outputs:

  • length(evec) results as:
    \text{ans} = 0

• Adding values to empty vectors can be done via concatenation:

  • Example operations:

  • evec = [evec 4] gives:
    \text{evec} = [4]

  • evec = [evec 11] results in:
    \text{evec} = [4, 11]

Removing Elements from Vectors and Matrices

• To delete an element in a vector:

  • Use indexing:

  • vec(3) = [] removes the element yielding:
    \text{vec} = [4, 5, 7, 8]

• Subsets can also be removed:

  • Example: vec(2:4) = [] eliminates specified slice.

• For matrices, removing individual elements is impossible; full rows or columns can be deleted:

  • Example: mat(:,2) = [] yields remaining matrix:
    \text{mat} = \begin{bmatrix} 7 & 8 \ 4 & 5 \end{bmatrix}

Three-Dimensional Matrices

• Example for creating a 3D matrix:

  • layerone = reshape(1:15, 3, 5) results in a 3x5 matrix:

  • layertwo can be created as:

  • layertwo = fliplr(flipud(layerone))

• Element-wise results can be arranged into a 3D array:

Mathematical Functions with Vectors and Matrices

• Finding the sine for every element:

  • Example:

  • vec = -2:1 gives the vector
    \text{sinvec} = sin(vec) yielding:
    \text{sinvec} = [-0.9093, -0.8415, 0, 0.8415]

• Sum Function: Sums elements of a vector, e.g. sum(vec1) returns: $\text{ans} = 15$.

• Product Function: Computes the product of all elements in vec2 yielding:

  • Example: prod(vec2) results in:
    $\text{ans} = 240$.

• Min and Max Functions:

  • Get minimum values using min(vec1), maximum with max(vec2), and both with indices:

• Example:

  • [a, b] = min([2, 4, -5, 7]), returns:

  • \text{a} = -5

  • \text{b} = 3

Working with Random Matrices

• Generate random matrices:

  • Example for random numbers: A=rand(2) gives an output:

Conclusion on Vectors and Matrices

• Understanding how to create, manipulate, and perform operations on vectors and matrices in MATLAB is key for advanced mathematical operations within the environment.