Linear algebra is the study of vectors and certain rules to manipulate vectors.
Vectors: are special objects that can be added together and multiplied by scalars to produce another object of the same kind.
Examples
Geometric vectors
Polynomials
Audio signals
Elements of $\Bbb{R}^n $(tuples of $n$ real numebers)
Geometric Interpretation of Systems of Linear Equatiion
In a system of linear equations
with two variables x1, x2, each linear equation defines a line on the x1x2-plane.
with three variables, x1, x2, x3, each linear equation defines a plane in three-dimensional space.
When we intersect these planes, i.e., satisfy all linear equations at the same time, we can obtain a solution set that is a plane, a line, a point or empty (when the planes have no common intersection).
Matrices
A central role in linear algebra
used to compactly represent systems of linear equations
represent linear functions (linear mappints)
Matrix
Matrix addtion
Matrix multiplication
Hadamard product: element-wise product. It is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimension as the operands, where each element i, j is the product of elements i, j of the original two matrices.
Matrix product
Identity matrix: $n × n$-matrix containing $1$ on the diagonal and $0$ everywhere else
