In Linear Algebra we study how to solve systems of linear equations along with vectors, vector spaces, matrices and linear transformations. A linear equation may have several variables but they always appear to the power of one. Thus $6x_1+5x_2\mathbin{-}x_3=10$ is a linear equation. When you solved simultaneous equations in high school you were in fact solving a system of two linear equations in two variables usually $x$ and $y$. In Linear algebra we often consider a system of $n$ linear equations in $n$ variables with special emphasis on the common case where $n=3$. You may be familiar with vectors from elementary physics where they are usually defined as quantities like velocity and force that have both magnitude and direction. In our work this picture will still be of use but we will also have a more abstract view where a vector is a collection of numbers which represents its components in some coordinate system. We will treat a vector space as a set of vectors that may be added together or scaled by a number so that the result is always in the original set. A matrix is a rectangular array of number. They are used in representing and solving large systems of linear equations efficiently. They also represent linear transformations like rotations and scaling. A linear transformation is a function that maps one vector space to another in such a way that it preserves that operations of vector addition and scalar multiplication. This last statement may be somewhat obscure. What do we mean by “preserves the operation of vector addition”. Think you two vector spaces $V_a$ and $V_b$. You can add two vectors in $V_a$ getting a vector that belongs to $V_a$ and then apply the linear transformation to move the result to $V_b$. Alternatively you first transform both vectors to $V_b$ and then add them there to get a vector in $V_b$. If the final vector is the same in both cases then the Linear Transformation preserves or respects the vector addition operation. Transformations or maps that preserve some structure when they operate on objects play a very important role in mathematics. Linear Algebra is very relevant to modern technology with applications in computer graphics, game and 3D rendering engines, data processing, neural networks and data optimization. They also have more traditional applications in physics and engineering in modelling systems and solving the resulting systems of equations.