This lecture is associated with the book Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares.
Focus on fundamental concepts regarding vectors.
Topics covered: notation, examples, basic operations, and special types of vectors.
Vectors and Notation
Definition
A vector is an ordered list of numbers.
Vectors can be written in two forms:
Vertically, as an array of numbers stacked on top of each other (column vector).
Inline, as a horizontal array.
Components and Dimension
Elements/Entries/Components/Coefs refer to the numbers in the vector.
Dimension/Length: number of elements in a vector.
Example: A vector with 4 elements is a 4-vector.
Entries are referred to by their position, e.g., the second element.
Scalars: The individual numbers within a vector, usually real numbers but can be complex.
Abstract Notation
Vectors are often denoted by symbols (lowercase a, uppercase X, etc.).
In various fields, vectors might be denoted with bold letters or arrows.
Entry Reference
Entry of a vector denoted by subscript notation, e.g., A<sub>i</sub> where i is the index.
Indices often denoted by i, j, k, etc.
Example: If A is a vector, A<sub>2</sub> refers to the second entry of A.
Equality of Vectors
Two vectors are equal if they have the same size and all corresponding elements are equal.
Symbol overloading: Using the same symbol (equality) for different contexts (e.g., numbers vs. vectors).
Special Vectors
Zero Vector
Denoted by 0.
Can also have a subscript to denote size, e.g., 0<sub>3</sub> is a zero vector of dimension 3.
Ones Vector
Denoted by bold 1, e.g., 1<sub>n</sub>.
A vector with all entries equal to 1.
Unit Vectors
Vectors with all entries 0 except one, which is 1.
Denoted by e<sub>i</sub>, where i is the position of the 1.
Example: e<sub>2</sub> in a 3-vector would be [0, 1, 0].
Block Vectors
Combining smaller vectors into a larger one by stacking them.
Notation: [b, c, d] represents stacking b, c, and d.
Example: s = [-1, 1], t = [0, 2, 2], combining them results in [s, t] = [-1, 1, 0, 2, 2].
Sparsity
A vector is sparse if many of its entries are 0.
Important for large dimensions (e.g., vectors with millions of entries).
Efficient storage and manipulation techniques available for sparse vectors.
NNZ: Number of non-zero elements in a vector.
Example: A zero vector is the sparsest with NNZ = 0.
Unit vector e<sub>17</sub> in a 100-vector has NNZ = 1.
Mathematical vs. Computer Language Notation
Standard mathematical notation vs. specific computer language syntax (e.g., Julia, Python).
Important to differentiate and not mix the two.
Learn both mathematical notation and the syntax of the chosen computer language.
Examples of possible syntax: ones(5) would create a '1'-vector of dimension 5 in a computer language.
Key Takeaway
This lecture introduces fundamental concepts and notations of vectors, laying the groundwork for more advanced topics and applications later in the course.