Linear algebra is the mathematics of vectors and the flat, straight-line structure of space. Where calculus studies things that curve and change, linear algebra studies things that are linear (that scale and add in simple, predictable ways) and that turns out to describe an astonishing amount of the world.
Two ways in
There are two doors into the subject, and they meet in the middle. One is systems of equations: two or three linear equations in two or three unknowns, the kind you solved in school. Linear algebra is what happens when there are hundreds of equations and hundreds of unknowns, and it gives a clean language for solving them all at once.
The other door is geometry: arrows in space that you can add together and stretch. These arrows are vectors, and the operations that move and reshape space in straight-line-preserving ways are linear transformations, captured by grids of numbers called matrices.
Why it is everywhere
Almost anything with many numbers working together speaks the language of linear algebra. A computer graphics engine rotates and scales a scene by multiplying points by matrices. Machine learning stores data as huge vectors and matrices and trains models by transforming them. Physics, economics, engineering, and statistics all reach for it whenever many quantities depend on many others in a straight-line way.
Where we are headed
This course builds the core in six steps. We start with vectors (the objects) and the ways to combine them. Then matrices, which are both tables of numbers and machines that act on vectors. Then the key idea that a matrix is a linear transformation of space. From there, the determinant measures how a transformation stretches area and tells us when it can be undone, and finally eigenvectors reveal the directions a transformation leaves pointing the same way. Keep the geometric picture in mind throughout (vectors as arrows, matrices as motions of space) and the algebra becomes something you can see.