Linear Algebra Fundamentals

Linear Algebra

• A function $f: E \rightarrow F$ is linear if $f(x+y) = f(x) + f(y)$ and $f(\lambda x) = \lambda f(x)$, for all $x, y \in E$ and $\lambda \in \mathbb{K}$.
• If $f \in \mathcal{L}(E, F)$ and $E$ is of dimension finite, then $\dim(E) = \dim(\operatorname{ker} f) + \dim(\operatorname{Im} f)$.
• $p \in \mathcal{L}(E)$ is a projector if and only if $p \circ p = p$.
• $u \in \mathcal{L}(E)$ is symmetric if $\langle u(x), y\rangle = \langle x, u(y)\rangle$, for all $x, y \in E$.
• Matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A = P^{-1} B P$.
• The determinant of matrix $A$ is defined as $\operatorname{det}(A) = \sum_{\sigma \in \mathcal{S}{n}} \epsilon(\sigma) \prod{i=1}^{n} a_{i, \sigma(i)}$.
• The cofactor of an element $a_{ij}$ is given by $C_{i j}=(-1)^{i+j} \operatorname{det}\left(M_{i j}\right)$, where $M_{i j}$ is the minor matrix.
• Matrix $A$ is invertible if and only if $\operatorname{det}(A) \neq 0$.
• The inverse of matrix $A$ is given by $A^{-1} = \frac{1}{\operatorname{det}(A)}^{t} \operatorname{Com}(A)$.
• Eigenvalues can be found using the characteristic equation $\operatorname{det}(A - \lambda I) = 0$.
• Matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$.
• Gauss reduction includes row exchange ($L_{i} \leftrightarrow L_{j}$), dilation ($L_{i} \leftarrow \lambda L_{i}$), and transvection ($L_{i} \leftarrow L_{i} + \lambda L_{j}$).
• Gauss pivot involves dividing the first row by $a_{11}$ to introduce a pivot of 1 and making zeros below

Group Theory

• A group is a set $G$ with a binary operation $\cdot: G \times G \rightarrow G$ that satisfies associativity, the existence of an identity element $e$, and inverses for each element.
• A group $(G, \cdot)$ is abelian if $a \cdot b = b \cdot a$ for all $a, b \in G$.
• $(\mathbb{Z}, +)$, $(\mathbb{Q}, +)$, $(\mathbb{R}, +)$, and $(\mathbb{C}, +)$ are abelian groups with the identity element 0 and the inverse $-a$.
• $(\mathbb{Q} \setminus {0}, \cdot)$, $(\mathbb{R} \setminus {0}, \cdot)$, and $(\mathbb{C} \setminus {0}, \cdot)$ are abelian groups with the identity element 1 and the inverse $a^{-1} = \frac{1}{a}$.
• $GL_n(\mathbb{R})$ is a group under matrix multiplication, but it's non-abelian for $n > 1$.
• $(\mathbb{Z}_n, +)$ is an abelian group where addition is performed modulo $n$.
• $(\mathbb{Z}_n^*, \cdot)$ is an abelian group where multiplication is performed modulo $n$ and elements are coprime to $n$.
• The order of $G$, denoted by $|G|$, is the number of elements in the group.
• $|\mathbb{Z}_n| = n$.
• $|\mathbb{Z}_p^*| = p - 1$ if $p$ is prime.
• The order of $a$, denoted by $|a|$, is the smallest positive integer $n$ such that $a^n = e$.
• In $(\mathbb{Z}_6, +)$, $|0| = 1$, $|1| = 6$, $|2| = 3$, and $|3| = 2$.
• If $a^m = e$, then $|a|$ divides $m$.
• For any element $a$ in a finite group $G$, $|a|$ divides $|G|$.
• A non-empty subset $H$ of $G$ is a subgroup of $G$ if $H$ is a group under the induced operation of $G$.
• $H$ is a subgroup of $G$ if and only if for all $a, b \in H$, $a \cdot b \in H$ and for all $a \in H$, $a^{-1} \in H$.
• For any group $G$, ${e}$ and $G$ are subgroups of $G$, where ${e}$ is the trivial subgroup.
• $n\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ for any $n \in \mathbb{Z}$.
• A non-empty subset $H$ of a finite group $G$ is a subgroup of $G$ if for all $a, b \in H$, $a \cdot b \in H$.

Chemical Kinetics

• Chemical kinetics (or reaction kinetics) is the study of reaction rates, how they are affected by reaction conditions, and reaction mechanisms.

Factors Affecting Reaction Rate

• Reactant concentration: Higher concentration means more frequent collisions, increasing the reaction rate.
• Temperature: Higher temperatures provide required activation energy for successful collisions, increasing the reaction rate.
• Physical state of reactants and surface area: Reactions occur at phase interfaces, and larger surface areas quicken the reaction.
• Solvent: Can change the reaction rates.
• Catalyst: Catalysts increase reaction rates without being permanently changed, by providing alternative reaction pathways with lower activation energy.

Reaction Rate

• Reaction rate is the changes in reactant or product concentration per unit time.

Rate Expression

• For the reaction $aA + bB \rightarrow cC + dD$, the rate expression is: $Rate = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$

Types of Rates

• Average rate: Calculated over long periods, $Average Rate = -\frac{\Delta[Reactant]}{\Delta t}$
• Instantaneous rate: The rate at a specific time, $Instantaneous Rate = \lim_{\Delta t \to 0} \frac{\Delta[Reactant]}{\Delta t}$
• Initial rate: Instantaneous rate at the start of the reaction ($t = 0$).

Rate Law

• The rate law shows the relationship between reaction rate and reactant concentrations.
• For the reaction $aA + bB \rightarrow cC + dD$, the rate law is $Rate = k[A]^m[B]^n$, where:
  • $k$ is the rate constant.
  • $m$ is the order of the reaction with respect to reactant A.
  • $n$ is the order of the reaction with respect to reactant B.
  • $m + n$ is the overall reaction order.
• Reaction order can be an integer, fraction, or zero and must be experimentally determined.

Determining Reaction Order

• Reaction order can be integer, fraction or zero. Reaction orders are determined using experimental data.

Integrated Rate Laws

• Integrated rate laws show reactant concentration as a function of time

First-Order Reactions

• The rate depends on the concentration of one reactant: $Rate = k[A]$
• The integrated rate law is: $ln[A]_t - ln[A]_0 = -kt$
  • $[A]_t$ is the concentration of A at time t.
  • $[A]_0$ is the initial concentration of A.

Second-Order Reactions

• The rate depends on the square of one reactant's concentration or product of the concentration of two reactants.
• $Rate = k[A]^2$ or $Rate = k[A][B]$
• The integrated rate law is: $\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt$

Zero-Order Reactions

• The rate is independent of the concentration of the reactant(s): $Rate = k$
• The integrated rate law is: $[A]_t - [A]_0 = -kt$

Half-Life

• Half-life ($t_{1/2}$) is the time it takes for the reactant concentration to decrease to half of its initial value.

Equations for Half-Life

• First-order: $t_{1/2} = \frac{0.693}{k}$
• Second-order: $t_{1/2} = \frac{1}{k[A]_0}$
• Zero-order: $t_{1/2} = \frac{[A]_0}{2k}$

Collision Theory

• Reactant molecules must collide with enough energy and appropriate orientation for a reaction to occur.

Activation Energy

• Activation energy ($E_a$) is the minimum energy needed for a reaction to occur.

Arrhenius Equation

• The Arrhenius equation expresses how temperature affects the rate constant: $k = Ae^{-\frac{E_a}{RT}}$
  • $k$ is the rate constant.
  • $A$ is the frequency factor.
  • $E_a$ is the activation energy.
  • $R$ is the gas constant ($8.314 J/(mol \cdot K)$).
  • $T$ is the absolute temperature (in Kelvin).

Reaction Mechanisms

• A reaction mechanism is a series of elementary steps showing the step-by-step process of how a chemical reaction occurs.

Elementary Steps

• Reactions that represents a single step of a reaction.

Rate - Determining Step

• The rate-determining step is the slowest elementary step that controls the overall reaction rate.

Intermediates

• Intermediates are species that are produced in one step and consumed in a later step in the reaction mechanism.

Catalysis

• Catalysts speed up reactions by providing an alternative pathway that has a lower activation energy.

Types of Catalysis

• Homogeneous catalysis: Catalysts are in the same phase as the reactants.
• Heterogeneous catalysis: Catalysts are in a different phase from the reactants.
• Enzyme catalysis: Enzymes (biological catalysts) catalyzing biochemical reactions.