Vector and Matrices
Vectors and Matrices
Vectors: The Building Blocks
At its core, a vector is just a list of data . Whether it’s a row of GDP numbers or a column of batting averages, vectors structure our data.
Visualizing Vectors
In (2D space), a vector is an arrow starting from the origin .
- Magnitude: The length of the arrow.
- Direction: Where the arrow points.
Understanding Matrices
A matrix is a rectangular array of numbers. It’s essentially a collection of vectors.
Types of Matrices
- Square Matrix: Same number of rows and columns ().
- Diagonal Matrix: Non-zero entries only on the diagonal.
- Scalar Matrix: A diagonal matrix where all diagonal elements are equal.
- Identity Matrix (): A scalar matrix with s on the diagonal.
Matrix Operations
- Addition: Add corresponding elements.
- Scalar Multiplication: Multiply every element by the scalar.
- Matrix Multiplication: The row of the first matrix dots with the column of the second.
Determinants
The determinant is a special number calculated from a square matrix.
- Invariance: Adding a multiple of one row to another does not change the determinant.
- Minors & Cofactors: Tools used to calculate determinants of higher-order matrices.
All Chapters in this Book
Vector and Matrices
Introduction to vectors, matrices, and their fundamental operations in linear algebra.
Solving Systems of Linear Equations
Mastering techniques to solve linear systems: Cramer's Rule, Inverse Matrix, and Gauss Elimination.
Introduction to Vector Space
Formal definition of vector spaces, axioms, and subspaces.
Basis and Dimension
Understanding the building blocks of vector spaces: Linear Independence, Spanning Sets, Basis, and Dimension.
Rank and Nullity
Exploring the fundamental subspaces of a matrix and the Rank-Nullity Theorem.
Linear Transformation
Mapping vector spaces: Homomorphisms, Isomorphisms, and Matrix Representations.
Equivalence and Similarity
Comparing matrices: When are two matrices really the same thing in disguise?
Affine Subspaces
Moving beyond the origin: Affine subspaces and mappings.
Inner Product Space
Geometry in vector spaces: Angles, Lengths, and Orthogonality.