Linear Algebra Explained with Simple Animations

A vector is a mathematical object that has both magnitude and direction.Each vector can be represented by its components in the x and y directions.Let's add another vector to see how vectors interact.To add vectors, we use the tip-to-tail method. Move the second vector so its start point touches the first vector's tip.The resultant vector goes from the start of the first vector to the tip of the second vector.Scalar multiplication stretches or shrinks a vector. When we multiply a vector by 2, it doubles in length while maintaining its direction.In physics, vectors are used to represent forces. Here's an object with two forces acting on it.The net force is the vector sum of all forces acting on the object.In video games, vectors control character movement. The velocity vector determines both speed and direction.As the character moves, its position is updated based on the velocity vector.A matrix is a powerful tool that can transform vectors and entire coordinate spaces.Let's see how matrices can transform shapes in space. We'll start with a simple square.A scaling matrix can stretch or compress space. This matrix will stretch horizontally and compress vertically.A rotation matrix can rotate the entire space. This matrix will rotate everything by forty-five degrees.A shear matrix creates a slanting effect, like pushing a rectangle into a parallelogram.Matrices can be combined to create complex transformations. This matrix combines scaling, rotation, and shearing all at once.In computer graphics, matrices are used to transform shapes and create animations.These transformations will help us understand eigenvalues and eigenvectors in our next section.When we apply a matrix transformation, most vectors change both their direction and magnitude.Watch how these vectors change direction when transformed by our matrix.However, some special vectors, called eigenvectors, maintain their direction under this transformation.When we apply the matrix transformation, these eigenvectors only change in length, not direction.The amount by which each eigenvector is stretched is called its eigenvalue.In data analysis, eigenvectors help us find the principal components - the directions of maximum variance in our data.In engineering, eigenvectors help identify natural vibration modes of structures.