In 2D, matrices can represent transformations that manipulate vectors. A vector can be thought of as a point in 2D space. When we multiply a vector by a matrix, we change its position based on the type of transformation represented by that matrix.
Consider the matrix for a 90-degree counterclockwise rotation:
\[ R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]
If we multiply a vector \( \mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix} \) by this matrix, we get a new vector \( \mathbf{v'} \) that is rotated 90 degrees.
Scaling matrix for scaling by a factor of 2:
\[ S = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \]
Multiplying a vector \( \mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix} \) by this matrix will double both its x and y components, effectively stretching the vector.
Understanding matrix transformations is crucial for computer graphics, physics, and machine learning. These transformations allow you to manipulate images, model 3D objects, and even classify data in high-dimensional spaces.