Lecture01: Fundamentals

Geometry of Linear Equations

Linear equations lay the foundation for the study of spaces, transformations, and matrices. They describe hyperplanes that may or may not intersect, and their graphical representation provides invaluable insight into the nature of systems of equations.

Systems of Linear Equations

A system of linear equations can have one solution, no solutions, or infinitely many solutions. Their geometric interpretation aids in understanding these possibilities.

Vectors

Vectors serve as the building blocks of linear algebra. They represent points in space and can be used to describe locations, directions, and much more.

Vector Addition and Scalar Multiplication

The arithmetic of vectors is both intuitive and geometrically interpretable. Vector addition corresponds to geometric translation, and scalar multiplication affects a vector’s magnitude without altering its direction.

Linear Equations Revisited

Linear equations, when viewed through the lens of vectors, provide a new perspective. Each equation can be associated with a vector, and their intersections can determine solutions.

Linear Combinations

Linear combinations offer a way to create new vectors by scaling and summing known vectors. The span of a set of vectors is the set of all vectors that can be produced through their linear combinations.

Dot Products

The dot product of two vectors provides insights into their relative orientations. It is closely related to the angle between the vectors and has pivotal significance in projections and orthogonality.

Dimensions

The concept of dimension extends beyond simple 2D and 3D spaces. In linear algebra, it refers to the number of vectors in a basis for a space, providing a measure of the “size” or “degree of freedom” within that space.