Quantum Mechanics

Quantum Mechanics

• Quantum mechanics studies matter and energy at the atomic and subatomic levels.

Key Concepts

• Quantization: Properties like energy and momentum come in discrete amounts.
• Wave-Particle Duality: Particles can act like waves, and waves can act like particles.
• Uncertainty Principle: Limits exist on how precisely position and momentum can be simultaneously known.

Mathematical Representation

Wave Function

• $\Psi(r, t)$ represents the state of a quantum system.
• $|\Psi(r, t)|^2$ indicates the probability density of finding a particle at position $r$ at time $t$.

Schrödinger Equation

• $i\hbar\frac{\partial}{\partial t}\Psi(r, t) = \hat{H}\Psi(r, t)$ describes quantum state change over time.
• $i$ is the imaginary unit.
• $\hbar$ is the reduced Planck constant.
• $\frac{\partial}{\partial t}$ is the partial derivative with respect to time.
• $\hat{H}$ is the Hamiltonian operator, representing the system's total energy.

Core Principles

Superposition

• $|\Psi\rangle = c_1|\Psi_1\rangle + c_2|\Psi_2\rangle$ explains that a quantum system can be in multiple states at once.
• $|\Psi\rangle$ is the state of the system.
• $|\Psi_1\rangle$ and $|\Psi_2\rangle$ are possible states.
• $c_1$ and $c_2$ are complex numbers indicating probability amplitudes for each state.

Measurement

• Measuring a quantum system forces it into a single, definite state.

Entanglement

• Two or more quantum particles become linked inseparably.

Applications

• Quantum Computing: Uses quantum mechanics for computation.
• Quantum Cryptography: Secure communication using quantum principles.
• Materials Science: Designing and understanding new materials.
• Medical Imaging: Enhancement of imaging techniques.

Key Figures

• Max Planck: Introduced quantization.
• Albert Einstein: Explained the photoelectric effect.
• Niels Bohr: Developed the atomic model.
• Werner Heisenberg: Formulated the uncertainty principle.
• Erwin Schrödinger: Developed the Schrödinger equation.
• Paul Dirac: Integrated quantum mechanics with relativity.

Thermodynamics

Energy

• The capacity to do work.

• Exists in thermal, mechanical, kinetic, potential, electrical, magnetic, chemical, and nuclear forms.

• Total energy: $E$ is the sum of all energy forms.

• Total energy on a unit mass basis: $e = E/m$.

• Total energy is divided into macroscopic and microscopic forms.

  Macroscopic Energy

  • Macroscopic energy is energy possessed by a system as a whole relative to an external reference frame.

  Kinetic Energy (KE)

  • $KE = \frac{1}{2}mV^2$ is energy due to motion relative to a reference frame.
  • Kinetic energy on a unit mass basis: $ke = \frac{1}{2}V^2$.

  Potential Energy (PE)

  • $PE = mgz$ is energy due to its elevation in a gravitational field.
  • Potential energy on a unit mass basis: $pe = gz$.

  Microscopic Energy

  • Microscopic energy is related to molecular structure and activity, independent of external reference frames.

  Internal Energy

  • $U$ is the sum of all microscopic energy forms.
  • Internal energy on a unit mass basis is denoted by $u$.
  • Internal energy $U$ comprises sensible, latent, chemical, and nuclear energies.
    • Sensible Energy: Portion associated with molecular kinetic energies.
    • Latent Energy: Internal energy linked to the system's phase.
    • Chemical Energy: Internal energy in atomic bonds within molecules.
    • Nuclear Energy: Energy within the atom's nucleus.
  • Thermodynamics focuses on changes in internal energy, $\Delta u$.

  Forms of Energy

  • Energy crosses boundaries of closed systems as heat and work.
  • Heat $Q$ is energy transferred due to temperature differences.
  • Work $W$ is energy transfer from a force acting through a distance.

• Heat and work are energy-transfer mechanisms.

The First Law of Thermodynamics

• Energy is neither created nor destroyed, only transformed.
• The change in a system's total energy equals the difference between energy entering and leaving: $E_{in} - E_{out} = \Delta E_{system}$.
• In rate form: $\dot{E}{in} - \dot{E}{out} = \frac{dE_{system}}{dt}$.
• Energy Balance: Net energy transfer equals the change in the system's energy content.
• For a closed system undergoing a cycle: $\Delta E_{system} = 0$.

Energy Transfer by Heat

• Heat is recognized only as it crosses boundaries.
• Adiabatic Process: No heat transfer occurs.
• Heat Transfer Mechanisms:
  • Conduction: Energy transfer via particle interactions.
  • Convection: Energy transfer between a surface and moving fluid.
  • Radiation: Energy emitted as electromagnetic waves.
• Heat added to a system is positive; heat removed is negative.

Energy Transfer by Work

• Work is energy transfer via force acting over a distance.
• Power: Work done per unit time.
• Work done by the system is positive; work done on the system is negative.

Mechanical Forms of Work

Moving Boundary Work (Pdv Work)

• $W_b = \int_{1}^{2} P dV$
• Area under the process curve on a P-V diagram represents boundary work.

Shaft Work

• $W_{sh} = T\theta$
• Power transmitted: $\dot{W}_{sh} = T\omega$. $\omega$ is angular velocity.

Spring Work

• $W_{spring} = \frac{1}{2}k(x_2^2 - x_1^2)$

Electrical Work

• $W_e = VIt$

Example: Shaft Work

• A 15-hp electric motor running at 1700 rpm applies a torque of 62.83 N.m.

  $$\begin{aligned} \dot{W} &= 15 \text{ hp} = 15 \text{ hp} \left( \frac{0.7457 \text{ kW}}{1 \text{ hp}} \right) = 11.1855 \text{ kW} \ \omega &= 1700 \text{ rpm} = 1700 \frac{\text{rev}}{\text{min}} \left( \frac{2\pi \text{ rad}}{1 \text{ rev}} \right) \left( \frac{1 \text{ min}}{60 \text{ s}} \right) = 178.02 \text{ rad/s} \ \dot{W}{sh} &= T\omega \ T &= \frac{\dot{W}{sh}}{\omega} = \frac{11.1855 \text{ kW}}{178.02 \text{ rad/s}} \left( \frac{1000 \text{ N.m/s}}{1 \text{ kW}} \right) = 62.83 \text{ N.m} \end{aligned}$$

The First Law of Thermodynamics

• For a closed system: $Q - W = \Delta E = \Delta U + \Delta KE + \Delta PE$
• $Q - W = m(u_2 - u_1) + \frac{1}{2}m(V_2^2 - V_1^2) + mg(z_2 - z_1)$

• For stationary closed systems: $Q - W = \Delta U = m(u_2 - u_1)$
  

Specific Heats

• The energy needed to raise the temperature of a unit mass of a substance by one degree.
• $c_v$: Specific heat at constant volume.
• $c_p$: Specific heat at constant pressure.
• Specific heat is a property, not a thermodynamic property.
• For ideal gases, $u$ depends only on $T$:

$$du = c_v(T)dT$$

$$\Delta u = u_2 - u_1 = \int_{1}^{2} c_v(T)dT$$

• For ideal gases, $h$ depends only on $T$:

$$dh = c_p(T)dT$$

$$\Delta h = h_2 - h_1 = \int_{1}^{2} c_p(T)dT$$

Constant Specific Heats

• At low temperatures, specific heats vary with temperature, but at high temperatures, this variation is small.

$$\Delta u = u_2 - u_1 \cong c_{v, avg}(T_2 - T_1)$$

$$\Delta h = h_2 - h_1 \cong c_{p, avg}(T_2 - T_1)$$

Internal Energy, Enthalpy, and Specific Heats of Ideal Gases

• Ideal Gas Equation: $P = \rho RT$
• Enthalpy: $h = u + Pv = u + RT$.
• $c_p$ and $c_v$ are related: $c_p = c_v + R \quad (kJ/kg.K)$
• Specific heat ratio: $k = \frac{c_p}{c_v}$

The First Law for Ideal Gases

Constant Volume Process

• $Q - W = m c_v (T_2 - T_1)$
• $W = 0$
• $Q = m c_v (T_2 - T_1)$

Constant Pressure Process

• $Q - W = m c_p (T_2 - T_1)$
• $W = P(V_2 - V_1)$
• $Q = m c_p (T_2 - T_1)$

Isothermal Process

• $Q - W = m c_v (T_2 - T_1)$
• $T_1 = T_2$
• $Q = W$
• $W = P_1V_1 \ln{\frac{V_2}{V_1}} = mRT_1 \ln{\frac{V_2}{V_1}}$

Polytropic Process

• $P V^n = C$
• $W = \frac{P_2V_2 - P_1V_1}{1-n}$
• $W = \frac{mR(T_2 - T_1)}{1-n}$

Isentropic Process

• $P V^k = C$
• $W = \frac{P_2V_2 - P_1V_1}{1-k}$
• $W = \frac{mR(T_2 - T_1)}{1-k}$

Example: Electric Heating of a Gas at Constant Pressure

• 3 kg of air at 20$^\circ$C and 200 kPa is heated until the volume doubles.
• The work done by the air is 252.15 kJ; the heat transfer is 883.3 kJ.

$$\begin{aligned} Q &= m c_p (T_2 - T_1) = (3 \text{ kg}) (1.005 \frac{\text{kJ}}{\text{kg.K}}) (586 \text{ K} - 293 \text{ K}) = 883.3 \text{ kJ} \ \frac{V_1}{T_1} &= \frac{V_2}{T_2} \ T_2 &= \frac{V_2}{V_1} T_1 = 2 T_1 = 2(293 \text{ K}) = 586 \text{ K} \ W &= P(V_2 - V_1) = P(2V_1 - V_1) = PV_1 = m R T_1 \ W &= (3 \text{ kg}) (0.287 \frac{\text{kJ}}{\text{kg.K}}) (293 \text{ K}) = 252.15 \text{ kJ} \end{aligned}$$

Lecture 24: Max Flow

Max-Flow Problem

Input

• $G = (V, E)$ is a directed graph.
• Each edge $e$ has capacity $c_e > 0$.
• Source $s$ and sink $t$ are special nodes.

Goal

• Find the maximum flow from $s$ to $t$.

Definition of a Flow

Definition

• $f: E \rightarrow R^{\geq 0}$ is a function.
• Capacity constraint: $f_e \leq c_e, \forall e \in E$
• Flow conservation: $\sum_{e \text{ into } u} f_e = \sum_{e \text{ out of } u} f_e$ for every node $u \neq s, t$.

Definition

• $|f| = \sum_{e \text{ out of } s} f_e$ is the value of a flow $f$, representing the amount of flow leaving $s$.

Ford-Fulkerson Algorithm

Key Idea

• Increase the flow by sending flow along an "augmenting path".

Definition

• Given a flow $f$, an augmenting path is a path from $s$ to $t$ in the residual graph $G_f$.

Definition

• The residual graph $G_f$ has the same vertices as $G$.
• Edges in $G_f$:
  • If $f_e < c_e$, there is an edge $e$ in $G_f$ with capacity $c_e - f_e$
  • If $f_e > 0$, there is an edge $e'$ in $G_f$ with capacity $f_e$ (reverse edge)

Ford-Fulkerson Algorithm

1. Start with $f_e = 0$ for all $e$
2. While there is an augmenting path $P$ in $G_f$:
   • Let $b = \min_{e \in P} $ capacity of $e$ in $G_f$
   • Augment flow along $P$ by $b$

Augmenting Flow

• For each edge $e$ in $P$:
• If $e$ is a forward edge: $f_e = f_e + b$
• If $e$ is a backward edge: $f_e = f_e - b$

Limitations and Considerations

• Update $G_f$ after augmenting flow.
• Each iteration takes $O(m)$ time.
• Integer capacities increase flow by at least 1 in each iteration.
• Max flow $\leq \sum_{e \text{ out of } s} c_e$.

Theorem

• Ford-Fulkerson finds the maximum flow in $G$.

Cuts in Graphs

Definition

• A cut $(A, B)$ of a graph $G = (V, E)$ is a partition of $V$ into two sets $A$ and $B$ such that $s \in A$ and $t \in B$.

Definition

• The capacity of a cut $(A, B)$ is the sum of the capacities of the edges from $A$ to $B$: $c(A, B) = \sum_{e \text{ out of } A} c_e$

Lemma

• For any flow $f$ and any cut $(A, B)$, the net flow across $(A, B)$ is equal to $|f|$: $\sum_{e \text{ out of } A} f_e - \sum_{e \text{ into } A} f_e = |f|$

Corollary

• For any flow $f$ and any cut $(A, B)$, $|f| \leq c(A, B)$.

Max-Flow Min-Cut Theorem

• The value of the maximum flow is equal to the capacity of the minimum cut.

Summary

• The Ford-Fulkerson algorithm determines the maximum flow.
• The max-flow min-cut theorem connects maximum flow to minimum cut.

Quick Start Guide to Geometry Dash

Goal

• Complete each level by jumping and flying through obstacles, timed to the music.

Controls

• Tap, click, press spacebar, or up arrow to jump.
• Hold to jump/fly continuously.
• Release to stop jumping/flying.

Icons

• Icons customizable with colors. Unlock via gameplay or purchase.

Game Modes

• Cube, Nave, Ball, UFO, and Wave modes offer distinct gameplay.
• Robot and Spider modes unlock as you progress.

Levels

• Includes 21 official levels plus thousands of player-created levels.

Practice

• Levels have a Practice Mode where checkpoints can be placed. Does not count toward completion.

Secrets

• Hidden secrets and rewards exist. Vaults unlocked with codes on titlescreen.

Secret Coins

• Hidden in most official levels; collecting unlocks special content.

Tips

• Time jumps to the music.
• Practice to refine your skills.
• Persevere.

Lecture 14: October 24

Vector Spaces

Definition

• A vector space is a set $V$ with:
  • An addition law: $+: V \times V \rightarrow V$
  • A scalar multiplication law: $\cdot : \mathbb{R} \times V \rightarrow V$

Properties

1. Associativity: $\mathbf{u}+(\mathbf{v}+\mathbf{w})=(\mathbf{u}+\mathbf{v})+\mathbf{w} \quad \forall \mathbf{u}, \mathbf{v}, \mathbf{w} \in V$
2. Commutativity: $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u} \quad \forall \mathbf{u}, \mathbf{v} \in V$
3. Neutral element: $\exists \mathbf{0} \in V$ such that $\mathbf{u}+\mathbf{0}=\mathbf{u} \quad \forall \mathbf{u} \in V$
4. Inverse element: $\forall \mathbf{u} \in V, \exists(-\mathbf{u}) \in V$ such that $\mathbf{u}+(-\mathbf{u})=\mathbf{0}$
5. Compatibility with scalar multiplication: $a \cdot(b \cdot \mathbf{u})=(a b) \cdot \mathbf{u} \quad \forall a, b \in \mathbb{R}, \mathbf{u} \in V$
6. Distributivity w.r.t. scalar addition: $(a+b) \cdot \mathbf{u}=a \cdot \mathbf{u}+b \cdot \mathbf{u} \quad \forall a, b \in \mathbb{R}, \mathbf{u} \in V$
7. Distributivity w.r.t. vector addition: $a \cdot(\mathbf{u}+\mathbf{v})=a \cdot \mathbf{u}+a \cdot \mathbf{v} \quad \forall a \in \mathbb{R}, \mathbf{u}, \mathbf{v} \in V$
8. Identity: $1 \cdot \mathbf{u}=\mathbf{u} \quad \forall \mathbf{u} \in V$

Remarks

• $a \mathbf{u}$ instead of $a \cdot \mathbf{u}$.
• The element $\mathbf{0}$ is unique.
• The inverse $-\mathbf{u}$ is unique.
• $0 \cdot \mathbf{u}=\mathbf{0}$ for all $\mathbf{u} \in V$.
• We also have $(-1) \cdot \mathbf{u}=-\mathbf{u}$.

Examples

1. $V=\mathbb{R}^{n}=\left{\left(\begin{array}{c}x_{1} \ \vdots \ x_{n}\end{array}\right): x_{1}, \ldots, x_{n} \in \mathbb{R}\right}$

   $\left(\begin{array}{c}x_{1} \ \vdots \ x_{n}\end{array}\right)+\left(\begin{array}{c}y_{1} \ \vdots \ y_{n}\end{array}\right)=\left(\begin{array}{c}x_{1}+y_{1} \ \vdots \ x_{n}+y_{n}\end{array}\right), \quad a\left(\begin{array}{c}x_{1} \ \vdots \ x_{n}\end{array}\right)=\left(\begin{array}{c}a x_{1} \ \vdots \ a x_{n}\end{array}\right)$

2. $V=\mathbb{R}^{\infty}=\left{\left(\begin{array}{c}x_{1} \ x_{2} \ \vdots\end{array}\right): x_{i} \in \mathbb{R}\right}$

   $\left(\begin{array}{c}x_{1} \ x_{2} \ \vdots\end{array}\right)+\left(\begin{array}{c}y_{1} \ y_{2} \ \vdots\end{array}\right)=\left(\begin{array}{c}x_{1}+y_{1} \ x_{2}+y_{2} \ \vdots\end{array}\right), \quad a\left(\begin{array}{c}x_{1} \ x_{2} \ \vdots\end{array}\right)=\left(\begin{array}{c}a x_{1} \ a x_{2} \ \vdots\end{array}\right)$

3. $V={$ Polynomials with real coefficients $}$

   $\left(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\right)+\left(b_{0}+b_{1} x+\cdots+b_{m} x^{m}\right)=\left(a_{0}+b_{0}\right)+\left(a_{1}+b_{1}\right) x+\cdots$

   $c\left(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\right)=c a_{0}+c a_{1} x+\cdots+c a_{n} x^{n}$

4. $V={$ Functions $f: \mathbb{R} \rightarrow \mathbb{R}}$

   $(f+g)(x)=f(x)+g(x), \quad(a f)(x)=a f(x)$

Subspaces

• A Subspace of a vector space $V$ is a subset $W \subset V$ that is also a vector space for the same operations $+, \cdot$

Theorem/Criterion

• $W \subset V$ is a subspace if and only if:
  1. $\mathbf{0} \in W$
  2. $\mathbf{u}, \mathbf{v} \in W \Rightarrow \mathbf{u}+\mathbf{v} \in W$
  3. $a \in \mathbb{R}, \mathbf{u} \in W \Rightarrow a \mathbf{u} \in W$

Remark

• Conditions (2) and (3) are equivalent to: $\mathbf{u}, \mathbf{v} \in W, a, b \in \mathbb{R} \Rightarrow a \mathbf{u}+b \mathbf{v} \in W$

Examples

1. $V=\mathbb{R}^{2}, \quad W=\left{\left(\begin{array}{l}x \ 0\end{array}\right): x \in \mathbb{R}\right}$.
2. $V=\mathbb{R}^{3}, \quad W=\left{\left(\begin{array}{l}x \ y \ 0\end{array}\right): x, y \in \mathbb{R}\right}$.
3. $V=\mathbb{R}^{3}, \quad W=\left{\left(\begin{array}{l}x \ x \ x\end{array}\right): x \in \mathbb{R}\right}$.
4. $V=\mathbb{R}^{3}, \quad W=\left{\left(\begin{array}{l}x \ y \ z\end{array}\right): x+y+z=0\right}$.
5. $V=\mathbb{R}^{3}, \quad W=\left{\left(\begin{array}{l}x \ y \ z\end{array}\right): x+y+z=1\right}$. $\mathbf{0} \notin W$ so $W$ is not a subspace.
6. $V={$ Functions $f: \mathbb{R} \rightarrow \mathbb{R}}, \quad W={$ Continuous functions $}$.
7. $V={$ Functions $f: \mathbb{R} \rightarrow \mathbb{R}}, \quad W={$ Differentiable functions $}$.
8. $V={$ Differentiable functions $}, \quad W={$ Functions $f$ such that $f^{\prime \prime}+f=0}$.

Linear Combinations

Definition

• Let $V$ be a vector space and $\mathbf{v}{1}, \ldots, \mathbf{v}{n} \in V$. A linear combination of $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ is a vector of the form $a_{1} \mathbf{v}{1}+\cdots+a{n} \mathbf{v}{n}$ for some $a{1}, \ldots, a_{n} \in \mathbb{R}$.

Definition

• The span of $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ is the set of all linear combinations of $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$: $\operatorname{Span}\left{\mathbf{v}{1}, \ldots, \mathbf{v}{n}\right}=\left{a_{1} \mathbf{v}{1}+\cdots+a{n} \mathbf{v}{n}: a{1}, \ldots, a_{n} \in \mathbb{R}\right}$

Theorem

$\operatorname{Span}\left{\mathbf{v}{1}, \ldots, \mathbf{v}{n}\right}$ is a subspace of $V$.

Definition

• We say that $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ span $V$ if $\operatorname{Span}\left{\mathbf{v}{1}, \ldots, \mathbf{v}{n}\right}=V$.

Examples

1. $V=\mathbb{R}^{3}, \quad \mathbf{v}{1}=\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{l}0 \ 1 \ 0\end{array}\right)$. Then $\operatorname{Span}\left{\mathbf{v}{1}, \mathbf{v}{2}\right}=\left{a\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right)+b\left(\begin{array}{l}0 \ 1 \ 0\end{array}\right): a, b \in \mathbb{R}\right}=\left{\left(\begin{array}{l}a \ b \ 0\end{array}\right): a, b \in \mathbb{R}\right}$.
2. $\mathbf{e}{1}=\left(\begin{array}{l}1 \ 0\end{array}\right), \mathbf{e}{2}=\left(\begin{array}{l}0 \ 1\end{array}\right) \operatorname{span} \mathbb{R}^{2}$. Indeed, $\left(\begin{array}{l}x \ y\end{array}\right)=x \mathbf{e}{1}+y \mathbf{e}{2}$.
3. $\mathbf{e}{1}=\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right), \mathbf{e}{2}=\left(\begin{array}{l}0 \ 1 \ 0\end{array}\right), \mathbf{e}{3}=\left(\begin{array}{l}0 \ 0 \ 1\end{array}\right) \operatorname{span} \mathbb{R}^{3}$. Indeed, $\left(\begin{array}{l}x \ y \ z\end{array}\right)=x \mathbf{e}{1}+y \mathbf{e}{2}+z \mathbf{e}{3}$.
4. $\mathbf{v}{1}=\left(\begin{array}{l}1 \ 1\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{c}1 \ -1\end{array}\right) \operatorname{span} \mathbb{R}^{2}$.
5. $\mathbf{v}{1}=\left(\begin{array}{l}1 \ 1\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{l}2 \ 2\end{array}\right)$ do not span $\mathbb{R}^{2}$. $\operatorname{Span}\left{\mathbf{v}{1}, \mathbf{v}{2}\right}$ is the line $y=x$, which is not equal to $\mathbb{R}^{2}$.
6. $V={$ Polynomials $}, \quad \mathbf{v}{1}=1, \mathbf{v}{2}=x, \mathbf{v}{3}=x^{2}, \ldots$ Then $\mathbf{v}{1}, \mathbf{v}{2}, \mathbf{v}{3}, \ldots$ span $V$.

Linear Independence

Definition

• Let $V$ be a vector space and $\mathbf{v}{1}, \ldots, \mathbf{v}{n} \in V$. We say that $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ are linearly independent if the only solution to the equation $a_{1} \mathbf{v}{1}+\cdots+a{n} \mathbf{v}{n}=\mathbf{0}$ is $a{1}=a_{2}=\cdots=a_{n}=0$. If $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ are not linearly independent, then we say that they are linearly dependent.

Examples

1. $\mathbf{e}{1}=\left(\begin{array}{l}1 \ 0\end{array}\right), \mathbf{e}{2}=\left(\begin{array}{l}0 \ 1\end{array}\right)$ are linearly independent in $\mathbb{R}^{2}$.
2. $\mathbf{e}{1}=\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right), \mathbf{e}{2}=\left(\begin{array}{l}0 \ 1 \ 0\end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{l}0 \ 0 \ 1\end{array}\right)$ are linearly independent in $\mathbb{R}^{3}$.
3. $\mathbf{v}{1}=\left(\begin{array}{l}1 \ 1\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{l}2 \ 2\end{array}\right)$ are linearly dependent in $\mathbb{R}^{2}$. Indeed, notice that $\mathbf{v}{2}=2 \mathbf{v}{1}$, so $-2 \mathbf{v}{1}+\mathbf{v}{2}=\mathbf{0}$.
4. $\mathbf{v}{1}=\left(\begin{array}{l}1 \ 1\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{c}1 \ -1\end{array}\right)$ are linearly independent in $\mathbb{R}^{2}$.
5. $\mathbf{v}{1}=\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{l}1 \ 1 \ 0\end{array}\right), \mathbf{v}_{3}=\left(\begin{array}{l}1 \ 1 \ 1\end{array}\right)$ are linearly independent in $\mathbb{R}^{3}$.
6. $V={$ Functions $f: \mathbb{R} \rightarrow \mathbb{R}}, \quad f(x)=x, g(x)=x^{2}$ $f$ and $g$ are linearly independent.

Theorem

• $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ are linearly dependent if and only if one of the vectors is a linear combination of the others.

Basis

Definition

• A basis of a vector space $V$ is a set of vectors $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ such that:

1. $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ are linearly independent.
2. $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ span $V$.

Examples

1. $\mathbf{e}{1}=\left(\begin{array}{l}1 \ 0\end{array}\right), \mathbf{e}{2}=\left(\begin{array}{l}0 \ 1\end{array}\right)$ is a basis of $\mathbb{R}^{2}$.
2. $\mathbf{e}{1}=\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right), \mathbf{e}{2}=\left(\begin{array}{l}0 \ 1 \ 0\end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{l}0 \ 0 \ 1\end{array}\right)$ is a basis of $\mathbb{R}^{3}$.
3. $\mathbf{v}{1}=\left(\begin{array}{l}1 \ 1\end{array}\right), \mathbf{v}{2}=\left(\begin{array}{c}1 \ -1\end{array}\right)$ is a basis of $\mathbb{R}^{2}$.
4. $V={$ Polynomials $}, \quad \mathbf{v}{1}=1, \mathbf{v}{2}=x, \mathbf{v}_{3}=x^{2}, \ldots$ is a basis of $V$.

Definition

• We say that $V$ is finite-dimensional if it has a basis with a finite number of vectors.

Theorem

• If $V$ is finite-dimensional, then all bases of $V$ have the same number of vectors.

Definition

• The dimension of a finite-dimensional vector space $V$ is the number of vectors in any basis of $V$.

Examples

1. $\operatorname{dim} \mathbb{R}^{2}=2$
2. $\operatorname{dim} \mathbb{R}^{3}=3$
3. $\operatorname{dim}{$ Polynomials of degree $\leq n}=n+1$ (basis: $1, x, x^{2}, \ldots, x^{n}$ )
4. $\mathbb{R}^{\infty}$ is not finite-dimensional.

Theorem

• Let $V$ be a vector space of dimension $n$.

1. If $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ are linearly independent, then they span $V$ (so they form a basis).
2. If $\mathbf{v}{1}, \ldots, \mathbf{v}{n}$ span $V$, then they are linearly independent (so they form a basis).
3. If $\mathbf{v}{1}, \ldots, \mathbf{v}{m}$ are linearly independent and $m