👉 The clear math behind linear algebra, a fundamental branch of mathematics, revolves around vectors and matrices as tools to represent and manipulate geometric objects and transformations. Vectors, denoted by arrows, can represent quantities with both magnitude (length) and direction, while matrices, composed of rows and columns of numbers, serve as compact representations of linear transformations—operations that change vectors without altering their direction or magnitude. The dot product, a scalar result from multiplying corresponding components of vectors, measures their similarity and is crucial for finding angles and projections. Matrix multiplication, though not commutative, combines linear transformations: multiplying a matrix by a vector applies the transformation to each component. Eigenvalues and eigenvectors, key concepts, reveal invariant directions (eigenvectors) scaled by a factor (eigenvalue) under transformation, simplifying complex operations. These principles underpin applications in physics, engineering, and data science, enabling efficient problem-solving through linear transformations and vector spaces.
