Vectors and Matrices

Vectors

• A vector represents an ordered list of numbers.
• A vector of size n is generally represented as a column vector with elements x₁, x₂, up to xₙ.
• Vectors can be represented as lists or NumPy arrays in Python.
• Vectors must be the same size to be added. Components are added element-wise.
• Scalar multiplication involves multiplying each vector element by the scalar value.

Matrices

• A matrix represents a rectangular array of numbers, organized into rows and columns.
• A matrix of size $m \times n$ has elements aᵢⱼ, where i is the row number and j is the column number.
• Matrices can be represented using NumPy arrays in Python.

Matrix Operations

• Matrix addition is performed element-wise on matrices of the same size. $$ A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \end{bmatrix} $$
• Scalar multiplication involves multiplying each matrix element by the scalar value. $$ \alpha A = \begin{bmatrix} \alpha a_{11} & \alpha a_{12} & \cdots & \alpha a_{1n} \ \alpha a_{21} & \alpha a_{22} & \cdots & \alpha a_{2n} \ \vdots & \vdots & \ddots & \vdots \ \alpha a_{m1} & \alpha a_{m2} & \cdots & \alpha a_{mn} \end{bmatrix} $$
• Matrix multiplication:
  • Two matrices A and B can be multiplied only if the number of columns in A equals the number of rows in B.
  • For an $m \times n$ matrix A and an $n \times p$ matrix B, the resulting matrix C ($C = AB$) has dimensions $m \times p$.
  • The element $c_{ij}$ of matrix C is computed as: $$ c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} $$