What is linear algebra, and when do I have to use it?

Answer 1

From the category point of view. Linear algebra is the study of the category of vector spaces and linear transformations. A vector space is an non-empty set such that if you add two vector, or multiply a vector by a scalar you have again a vector. A linear transformation is a map between vector spaces that preserves the structure that is f(v+w)=f(v)+f(w) and f(cv)=cf(v). So, when you study linear algebra you are dealing with the properties of vectors (linear independence, linear combinations, basis, dimension, spanned spaces) and properties of linear transformations (kernel, image, dimension theorem) that makes this category consistent.

If you want an answer from the point of view of applications, any linear transformation between linear spaces of finite dimension is represented by a matrix. Matrices have many applications in calculus, linear programming, differential equations, economics, social sciences, medicine, and more.