1. Let's start by stating the problem: explaining what linear algebra is in a step-by-step manner. 2. Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. 3. A vector is an object that has both magnitude and direction, and can be represented as an ordered list of numbers, for example, $\mathbf{v} = (v_1, v_2, \ldots, v_n)$. 4. Vector spaces are collections of vectors that can be added together and multiplied by scalars (numbers), satisfying certain rules like associativity, commutativity, and distributivity. 5. Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication, meaning they map lines to lines or points. 6. Systems of linear equations are sets of equations where each equation is linear, and linear algebra provides methods to solve these systems efficiently using matrices and determinants. 7. Matrices are rectangular arrays of numbers that represent linear transformations or systems of equations. 8. Key operations in linear algebra include matrix addition, multiplication, finding inverses, determinants, eigenvalues, and eigenvectors. 9. Linear algebra is fundamental in many fields such as computer graphics, engineering, physics, machine learning, and more because it provides tools to model and solve real-world problems involving linear relationships. 10. In summary, linear algebra studies vectors and linear mappings between vector spaces, providing a framework to solve linear systems and understand geometric transformations. Final answer: Linear algebra is the study of vectors, vector spaces, linear transformations, and systems of linear equations, providing essential tools for solving problems involving linear relationships.