Functions Between Vector Spaces

Linear Algebra (Matrices) · Axiom Academy

Functions Between Vector Spaces Before linear transformations get their definition, meet the bigger idea: a transformation is just a function — it takes a whole space and sends every vector in it somewhere. A transformation acts on an entire space at once You already know a function takes a number and returns a number. A transformation does the same thing for vectors : it takes a vector as input and returns a vector as output. The surprise is the scale of it — a single transformation T doesn't move one vector, it moves every vector in the space at the same time. Watch a grid of vectors living in the input space (left) get carried, all together, into the output space (right) under a sample transformation T . Each gridline slides to where T sends it. Notice the grid stays an orderly grid — a hint of a special structure we'll name in the next lesson. One rule, applied everywhere: that's what makes it a transformation of the whole space, not just of a single vector. Pick an input — it has exactly one output Drag the blue arrow to choose any input vector in V . The orange arrow shows , its image in W . The point of a function: for each input you choose, there is one and only one output. Move around and watch the single output track it — never two outputs, never none. (Here T multiplies the input by a fixed matrix; the exact rule comes later.) Same input, same output, every time — that is exactly what "well-defined function" means. Every input has a home in the output space

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.