Vectors and Matrices

Vectors

• A vector possesses direction, sense, and a norm (length).
• Vectors can be represented as arrows, a pair of points, or a column matrix.
• An element of a vector space.

Vector Operations

• Vector addition is denoted as $\vec{u} + \vec{v}$.
• Scalar multiplication is denoted as $k \cdot \vec{u}$.
• The dot product is calculated as $\vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot cos(\theta)$.
• The cross product, applicable in 3D, is denoted as $\vec{u} \times \vec{v}$.

Matrices

• A matrix is a table of numbers arranged in rows and columns.
• The element at row i and column j is denoted as $a_{ij}$.
• $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \ a_{21} & a_{22} &... & a_{2n} \... &... &... &... \ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$ represents a matrix with m rows and n columns.

Matrix Operations

• Matrix addition $A + B$ requires both matrices to have the same dimensions.
• Scalar multiplication is written as $k \cdot A$.
• Matrix multiplication $A \cdot B$ requires the number of columns in A to equal the number of rows in B.
• Transposition $A^T$ swaps rows and columns.

Matrix Types

• Square matrix: has an equal number of rows and columns ($m = n$).
• Identity matrix: Denoted as $I = \begin{bmatrix} 1 & 0 &... & 0 \ 0 & 1 &... & 0 \... &... &... &... \ 0 & 0 &... & 1 \end{bmatrix}$.
• Diagonal matrix: Elements $a_{ij} = 0$ if $i \neq j$.
• Symmetric matrix: Satisfies the condition $A^T = A$.
• Triangular matrix: All elements above (lower triangular) or below (upper triangular) the diagonal are zero.

Linear Equation Systems

• A system of linear equations can be represented in matrix form as $A \cdot \vec{x} = \vec{b}$.
• In this representation:
  • $A$ is the coefficient matrix.
  • $\vec{x}$ is the vector of unknowns.
  • $\vec{b}$ is the vector of constants.

Methods to Solve Linear Equation Systems

• Gauss Method
• Gauss-Jordan Method
• Cramer's Rule (applicable if $A$ is square and invertible)

Determinant

• The determinant of a square matrix $A$, denoted as $det(A)$ or $|A|$, is a scalar.
• If $det(A) \neq 0$, $A$ is invertible.
• If $det(A) = 0$, $A$ is not invertible.

Inverse

• The inverse of a square matrix $A$, denoted as $A^{-1}$, satisfies $A \cdot A^{-1} = A^{-1} \cdot A = I$.

Vector Spaces

• A set with two operations:
  • Addition: $\vec{u} + \vec{v}$.
  • Multiplication by a scalar: $k \cdot \vec{u}$.
• These operations must satisfy associativity, commutativity, identity element, inverse element properties, etc.

Vector Space Examples

• $\mathbb{R}^n$: set of vectors with $n$ real components.
• $\mathbb{C}^n$: set of vectors with $n$ complex components.
• The set of $m \times n$ matrices.
• The set of continuous functions over an interval.

Subspaces

• Defined as a subset of a vector space that is also a vector space.

Basis

• A set of linearly independent vectors that span the vector space.

Dimension

• The number of vectors in a basis of a vector space.

Linear Transformations

• A function between two vector spaces that preserves addition and scalar multiplication.
• $T: V \rightarrow W$, where $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$, and $T(k \cdot \vec{u}) = k \cdot T(\vec{u})$.

Matrix Representation

• A linear transformation can be represented as $T(\vec{x}) = A \cdot \vec{x}$, where $A$ is the linear transformation matrix.

Eigenvalues and Eigenvectors

• $\vec{v}$ is an eigenvector of $A$ if $A \cdot \vec{v} = \lambda \cdot \vec{v}$, where $\lambda$ is an eigenvalue of $A$.

Momentum

• Momentum ($\vec{p}$) is defined as the product of mass ($m$) and velocity ($\vec{v}$): $\vec{p} = m \vec{v}$.
• It is a vector quantity with the unit kg m/s.

Impulse

• Impulse ($\vec{J}$) is the change in momentum of an object when a force acts on it over a time interval ($\Delta t$): $\vec{J} = \vec{F}_{net} \Delta t$.
• Impulse is a vector that shares the same direction as the net force.

Impulse-Momentum Theorem

• Impulse equals the change in momentum: $\vec{F}{net} \Delta t = \Delta \vec{p} = \vec{p}{f} - \vec{p}_{i}$.

Conservation of Momentum

• If the net external force on a system is zero, the total momentum is conserved: $\vec{p}{i} = \vec{p}{f}$.
• For two interacting particles (1 and 2) with no net external force: $\vec{p}{1i} + \vec{p}{2i} = \vec{p}{1f} + \vec{p}{2f}$.

Collisions

• An elastic collision conserves kinetic energy.
• An inelastic collision does not conserve kinetic energy; some kinetic energy is converted to other forms.
• Note: Momentum is conserved in both elastic and inelastic collisions as long as there is no net external force.

One Dimensional Collisions

• For one-dimensional collisions: $p_{1i} + p_{2i} = p_{1f} + p_{2f}$.
• For perfectly inelastic collisions (objects stick together): $m_{1}v_{1i} + m_{2}v_{2i} = (m_{1} + m_{2})v_{f}$.