Vectors and matrices

Vectors and Matrices are fundamental concepts in linear algebra that are widely used in mathematics, physics, computer science, and engineering.

Vectors

• Definition: A vector is an ordered list of numbers, which can represent physical quantities like displacement, velocity, or force. In an $n$-dimensional space, a vector is typically represented as $\mathbf{v} = [v_1, v_2, \ldots, v_n]$.
• Properties: Vectors have magnitude and direction. They can be added together and multiplied by scalars (real numbers).
• Types: Vectors can be classified as column vectors (vertical) or row vectors (horizontal).

Matrices

• Definition: A matrix is a rectangular array of numbers arranged in rows and columns. For example, a $m \times n$ matrix has $m$ rows and $n$ columns, represented as $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$.
• Operations: Matrices can be added, multiplied, and transposed. They can also represent linear transformations.

Applications

• Systems of Equations: Both vectors and matrices are used to represent and solve systems of linear equations.
• Computer Graphics: They are essential for transformations and rendering.
• Data Science: Matrices are often used to store and manipulate datasets in machine learning algorithms.

In summary, vectors are one-dimensional arrays that represent quantities with direction and magnitude, while matrices are two-dimensional arrays used for various operations and transformations in linear systems.