What Are Linear Transformations?

Linear transformations are special types of functions that connect two vector spaces. They help keep the rules of adding vectors and multiplying them by numbers the same.

When we say a function $T: V \to W$ (where $V$ and $W$ are vector spaces) is linear, it follows these two main rules:

1. Additivity: If you take two vectors, $\mathbf{u}$ and $\mathbf{v}$, from space $V$ and add them together, the result from the function $T$ should be the same as first applying $T$ to each vector and then adding the results.
   • So: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. Homogeneity: If you multiply a vector $\mathbf{u}$ from space $V$ by a number $c$, then applying the function $T$ should give you the same result as first applying $T$ to the vector and then multiplying the result by that number.
   • So: $T(c\mathbf{u}) = cT(\mathbf{u})$

Why Are Linear Transformations Important?

• Keeping Dimensions: Linear transformations can connect spaces that have different sizes or dimensions. This shows how they change or transform those spaces.

• Using Matrices: Every linear transformation can be shown using a matrix (a grid of numbers). This makes it easier to do calculations.

• Real-World Uses: They are very useful in solving math problems like systems of linear equations. They also help us understand important concepts like eigenvalues and eigenvectors. Additionally, they play a big role in computer graphics and other applications.