Hoja de repaso: Fundamentals of Linear Algebra and Complex Numbers

📋 Course Outline

1. Complex Number Parts & Functions
2. Binary Operations & Properties
3. Relations & Image Types
4. Uniqueness of Identity & Inverses
5. Linear Subspaces & Conditions
6. Basis & Dimension Theory
7. Sum & Direct Sum of Spaces
8. Rank & Maximal Independent Subset
9. Linear Transformations & Isomorphisms
10. Dual Space & Dual Basis
11. Matrix Representation & Operations
12. Kernel, Image & Rank of Operators

📖 1. Complex Number Parts & Functions

🔑 Key Concepts & Definitions

• Complex Number: A number of the form $z = a + bi$, where $a, b \in \mathbb{R}$ and $i^2 = -1$.

• Real Part ($\Re(z)$): The component $a$ of the complex number $z = a + bi$.

• Imaginary Part ($\Im(z)$): The component $b$ of the complex number $z = a + bi$.

• Modulus ($|z|$): The distance of $z$ from the origin in the complex plane, defined as $|z| = \sqrt{a^2 + b^2}$.

• Argument ($\arg(z)$): The angle $\theta$ between the positive real axis and the line segment from the origin to $z$, typically in $[-\pi, \pi)$.

• Trigonometrical (Polar) Form: Representation of $z$ as $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$.

📝 Essential Points

• The real part and imaginary part are extracted directly from the algebraic form: $z = a + bi$.

• The modulus relates to the magnitude of the complex number and is used in the polar form.

• The argument can be computed using $\arg(z) = \arctan(\frac{b}{a})$, considering the quadrant for correct angle determination.

• Operations such as multiplication and division are simplified in polar form:

  • $z_1 z_2 = |z_1||z_2| (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))$.
  • $\frac{z_1}{z_2} = \frac{|z_1|}{|z_2|} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$.

• De Moivre's Theorem: For integer $n$, $(r (\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta).$ It facilitates raising complex numbers to powers and extracting roots.

💡 Key Takeaway

Understanding the parts of complex numbers and their functions allows for efficient manipulation in algebraic and geometric contexts, especially through polar representation and De Moivre's theorem, which simplifies powers and roots of complex numbers.

📖 2. Binary Operations & Properties

🔑 Key Concepts & Definitions

• Binary Operation: A rule that combines any two elements of a set to produce another element within the same set. Formally, a function $* : S \times S \to S$.

• Associativity: A property of a binary operation $*$ where for all $a, b, c \in S$, $(a * b) * c = a * (b * c)$.

• Commutativity: A property where for all $a, b \in S$, $a * b = b * a$.

• Identity Element (Neutral Element): An element $e \in S$ such that for all $a \in S$, $a * e = e * a = a$.

• Inverse Element: For $a \in S$, an element $a^{-1}$ such that $a * a^{-1} = a^{-1} * a = e$, where $e$ is the identity element.

• Group: A set $G$ with a binary operation $*$ satisfying closure, associativity, existence of an identity element, and inverses for all elements.

📝 Essential Points

• Properties Determine Structure: The combination of properties like associativity, commutativity, identity, and inverses define algebraic structures such as groups, rings, and fields.

• Associativity and Commutativity: Not all binary operations are associative or commutative; these properties are crucial for defining specific algebraic systems.

• Identity and Inverses: The existence of an identity element and inverses allows for solving equations within the set, enabling the structure of groups.

• Subsets and Substructures: Subsets that are closed under the operation and contain the identity and inverses form subgroups, subrings, etc.

• Operations on Complex Numbers: Examples include addition and multiplication, which are associative and have identities; multiplication is also commutative in the field of complex numbers.

• Properties in Linear Algebra: Addition of vectors is associative and commutative, with the zero vector as the identity; scalar multiplication distributes over vector addition.

💡 Key Takeaway

Binary operations form the foundation of algebraic structures; their properties such as associativity, commutativity, and the existence of identity and inverse elements are essential for defining and understanding groups, rings, and fields.

📖 3. Relations & Image Types

🔑 Key Concepts & Definitions

• Relation: A rule that associates elements of one set with elements of another set (or the same set). Formally, a subset of the Cartesian product of two sets.
• Reflexive Relation: A relation where every element is related to itself.
• Symmetric Relation: A relation where if an element A is related to B, then B is related to A.
• Transitive Relation: A relation where if A is related to B and B is related to C, then A is related to C.
• Injective (One-to-One) Function: A function where each element of the domain maps to a unique element of the codomain; no two distinct inputs map to the same output.
• Surjective (Onto) Function: A function where every element of the codomain has at least one pre-image in the domain.
• Bijective Function: A function that is both injective and surjective; a perfect pairing between domain and codomain.

📝 Essential Points

• Relations can be characterized by properties like reflexivity, symmetry, and transitivity, which define their structure and behavior.
• Functions (images) are classified based on injectivity, surjectivity, and bijectivity, affecting their invertibility and the nature of their images.
• The image (or range) of a linear or set-theoretic relation determines the subset of the codomain that is actually "hit" by the relation.
• Understanding the properties of relations helps in defining equivalence relations, partial orders, and functions' invertibility.
• In the context of linear algebra, the image of a linear transformation is a subspace of the codomain, with properties linked to the rank and nullity.

💡 Key Takeaway

Relations and images are fundamental in understanding how elements are connected within sets and how functions map elements, with properties like symmetry, transitivity, and injectivity shaping their structure and applications in mathematics.

📖 4. Uniqueness of Identity & Inverses

🔑 Key Concepts & Definitions

• Identity Element: An element $e$ in a set $G$ with a binary operation $*$ such that for all $a \in G$, $e * a = a * e = a$.

• Inverse Element: For each $a \in G$, an element $a^{-1}$ in $G$ satisfying $a * a^{-1} = a^{-1} * a = e$.

• Uniqueness of Identity: The identity element in a set with a binary operation, if it exists, is unique.

• Existence of Inverses: An element $a$ in a set $G$ has an inverse if there exists $a^{-1}$ such that $a * a^{-1} = a^{-1} * a = e$.

• Uniqueness of Inverses: If an element $a$ has two inverses $b$ and $c$, then $b = c$.

📝 Essential Points

• Uniqueness of Identity:

  • If two elements $e$ and $e'$ both serve as identities, then $e = e'$.
  • Proof: $e * e' = e'$ (since $e$ is identity), and $e * e' = e$ (since $e'$ is identity), hence $e = e'$.

• Existence and Uniqueness of Inverses:

  • If $a$ has two inverses $b$ and $c$, then $b = c$.
  • Proof: $b = b * e = b * (a * c) = (b * a) * c = e * c = c$.

• Implication in Groups:

  • In a group, each element has a unique inverse, and the identity element is unique.

• Associativity (not necessarily assumed here, but crucial in groups):

  • Ensures the well-definedness of inverse operations and the structure's properties.

💡 Key Takeaway

The identity element and inverses in a set with a binary operation are both unique when they exist, forming the foundation for algebraic structures like groups where these properties are essential for consistency and structure.

📖 5. Linear Subspaces & Conditions

🔑 Key Concepts & Definitions

• Linear Subspace: A subset $W$ of a vector space $V$ that is closed under vector addition and scalar multiplication. Formally, if $u, v \in W$ and $\alpha \in \mathbb{F}$, then $u + v \in W$ and $\alpha u \in W$.

• Linear Span (Linear Hull): The smallest linear subspace containing a given set $S \subseteq V$. It consists of all finite linear combinations of vectors in $S$.

• Linear Dependence & Independence:

  • Dependent: Vectors $v_1, \dots, v_k$ are linearly dependent if there exist scalars, not all zero, such that $\alpha_1 v_1 + \dots + \alpha_k v_k = 0$.
  • Independent: Vectors are linearly independent if the only scalars satisfying the above are all zero.

• Subspace Criteria: A subset $W \subseteq V$ is a subspace if:

  1. Zero vector $0 \in W$.
  2. Closed under addition: $u, v \in W \Rightarrow u + v \in W$.
  3. Closed under scalar multiplication: $v \in W, \alpha \in \mathbb{F} \Rightarrow \alpha v \in W$.

• Subspace Conditions: For a subset $W \subseteq V$, to verify $W$ is a subspace, check the above three conditions.

📝 Essential Points

• Subspace Verification: To prove $W$ is a subspace, verify the three conditions directly or use known subspaces (e.g., kernel of a linear transformation, span of vectors).

• Kernel and Image:

  • Kernel: The set of vectors mapped to zero by a linear transformation $T: V \to U$, i.e., $\ker(T) = \{ v \in V : T(v) = 0 \}$. It is always a subspace.
  • Image: The set $\operatorname{Im}(T) = T(V)$, which is also a subspace of $U$.

• Linear Dependence & Basis:

  • A basis of a subspace $W$ is a linearly independent set that spans $W$.
  • The dimension of $W$ is the number of vectors in any basis.

• Conditions for Subspace:

  • The intersection of subspaces is a subspace.
  • The sum of subspaces $W_1 + W_2$ (set of all sums $w_1 + w_2$) is a subspace.

• Linear Subspace of $V$: Any non-empty subset closed under linear combinations is a subspace.

💡 Key Takeaway

A linear subspace is a fundamental structure in linear algebra characterized by closure under addition and scalar multiplication. Verifying subspace properties involves checking these closure conditions, with kernel and image of linear transformations serving as canonical examples. Understanding subspaces is essential for analyzing the structure of vector spaces, solving systems, and studying linear transformations.

📖 6. Basis & Dimension Theory

🔑 Key Concepts & Definitions

• Vector Space: A set $V$ over a field $F$ with two operations (vector addition and scalar multiplication) satisfying axioms like associativity, commutativity, existence of additive identity and inverses, distributivity, etc.

• Basis: A minimal set of vectors $\{v_1, v_2, ..., v_n\}$ in $V$ such that every vector in $V$ can be expressed uniquely as a linear combination of these vectors.

• Dimension: The number of vectors in any basis of $V$. For finite-dimensional spaces, all bases have the same number of elements.

• Linear Independence: A set of vectors $\{v_1, v_2, ..., v_k\}$ where no vector can be expressed as a linear combination of the others.

• Linear Dependence: A set of vectors where at least one vector can be written as a linear combination of the others.

• Span: The set of all linear combinations of a given set of vectors, denoted as $\text{span}\{v_1, v_2, ..., v_k\}$.

• Subspace: A subset $W \subseteq V$ that is closed under addition and scalar multiplication, and contains the zero vector.

📝 Essential Points

• Any finite-dimensional vector space $V$ has a basis with exactly $\dim V$ vectors.

• The basis provides a coordinate system for the space; every vector has a unique coordinate representation relative to the basis.

• The dimension is an invariant; it does not depend on the choice of basis.

• The size of a basis is minimal for spanning the space, and any spanning set with $\dim V$ vectors contains a basis.

• Subspaces can be extended to bases of the entire space, and bases of subspaces can be extended to bases of the whole space.

• The number of vectors in different bases of the same space is always equal, confirming the concept of dimension.

• The linear dependence or independence of vectors determines whether they can form a basis or need to be extended.

💡 Key Takeaway

A basis is a minimal, linearly independent set that spans the entire vector space, and the number of vectors in any basis defines the space's dimension, serving as a fundamental measure of its size and structure.

📖 7. Sum & Direct Sum of Spaces

🔑 Key Concepts & Definitions

• Sum of Subspaces: Given subspaces $U, V$ of a vector space $W$, their sum $U + V$ is the set of all vectors that can be written as $u + v$, where $u \in U$, $v \in V$. Formally: $U + V = \{ u + v \mid u \in U, v \in V \}$ It is a subspace of $W$.

• Direct Sum of Subspaces: The sum $U + V$ is called a direct sum, denoted $U \oplus V$, if every vector in $U + V$ can be uniquely written as $u + v$. Equivalently: $U \cap V = \{ 0 \}$ and $U + V$ is the internal direct sum.

• Internal vs External Direct Sum:

  • Internal: The subspaces are within the same space $W$, and their sum is direct.
  • External: Constructed as a new space from two spaces $U$ and $V$, with the direct sum being their external sum.

• Dimension Formula for Sum: $\dim(U + V) = \dim(U) + \dim(V) - \dim(U \cap V)$ For the direct sum, since $U \cap V = \{ 0 \}$, this simplifies to: $\dim(U \oplus V) = \dim(U) + \dim(V)$

📝 Essential Points

• The sum $U + V$ is always a subspace, but it is a direct sum only if $U \cap V = \{ 0 \}$.
• The uniqueness of representation in a direct sum is crucial: each vector in $U \oplus V$ has exactly one decomposition as $u + v$.
• The dimension of the sum relates to the dimensions of the subspaces and their intersection; understanding this helps in decomposing spaces.
• The concept extends to finite and infinite-dimensional spaces, with the same fundamental properties.
• The decomposition theorem: Any finite-dimensional space $W$ can be expressed as a direct sum of subspaces, often a basis is extended to a direct sum decomposition.

💡 Key Takeaway

The sum of subspaces combines their elements, but the direct sum ensures a unique decomposition of vectors, which is fundamental for analyzing the structure of vector spaces and their subspaces. The dimension formula links the sizes of the subspaces and their intersection, providing a tool for space decomposition.

📖 8. Rank & Maximal Independent Subset

🔑 Key Concepts & Definitions

• Rank of a system of vectors: The maximum number of linearly independent vectors in the system. It equals the dimension of the subspace spanned by those vectors.

• Linearly independent set: A set of vectors where no vector can be expressed as a linear combination of the others.

• Maximal independent subset: The largest possible subset of vectors within a set that remains linearly independent; its size equals the rank of the original set.

• Linear span: The set of all linear combinations of a given set of vectors, forming a subspace.

• Dimension of a subspace: The number of vectors in its basis, i.e., the size of its maximal linearly independent subset.

• Properties:

  • The rank of a set of vectors is less than or equal to the number of vectors in the set.
  • Any linearly independent subset can be extended to a basis of the span.
  • The rank is invariant under linear transformations that preserve independence.

📝 Essential Points

• The rank provides a measure of the "independent information" in a set of vectors.
• To determine the rank, identify the largest subset of vectors that are linearly independent.
• The maximal independent subset is crucial for forming bases and understanding the structure of subspaces.
• The rank of a matrix (formed by vectors as columns or rows) equals the dimension of the image of the associated linear transformation.
• The rank-nullity theorem relates the rank of a linear transformation to the dimension of its kernel (nullity):
  $\text{dim(domain)} = \text{rank} + \text{nullity}$
• In practical terms, reducing a matrix to row echelon form helps identify the rank by counting non-zero rows.

💡 Key Takeaway

The rank of a set of vectors indicates the maximum number of linearly independent vectors it contains, serving as a fundamental measure of the set's capacity to generate subspaces and determine linear dependence or independence.

📖 9. Linear Transformations & Isomorphisms

🔑 Key Concepts & Definitions

• Linear Transformation: A function $T: V \to W$ between vector spaces over the same field such that for all $u, v \in V$ and scalars $a, b$: $T(au + bv) = aT(u) + bT(v)$
• Kernel (Null Space): The set of vectors in $V$ mapped to the zero vector in $W$: $\ker(T) = \{ v \in V \mid T(v) = 0 \}$
• Image (Range): The set of vectors in $W$ that are images of vectors in $V$: $\operatorname{Im}(T) = \{ T(v) \mid v \in V \}$
• Isomorphism: A bijective linear transformation $T: V \to W$ that preserves vector space structure; i.e., $T$ is linear, one-to-one, and onto.
• Linear Operator: A linear transformation from a vector space to itself, $T: V \to V$.
• Rank: The dimension of the image of a linear transformation: $\operatorname{rank}(T) = \dim(\operatorname{Im}(T))$
• Nullity: The dimension of the kernel: $\operatorname{nullity}(T) = \dim(\ker(T))$
• Fundamental Theorem of Linear Algebra: For a linear transformation $T: V \to W$: $\dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)$

📝 Essential Points

• Linearity: Ensures the transformation respects vector addition and scalar multiplication.
• Kernel and Image: Key to understanding the structure of $T$; kernel measures injectivity, image measures surjectivity.
• Isomorphisms: When $T$ is bijective, $V$ and $W$ are structurally identical (isomorphic). This implies equal dimensions.
• Matrix Representation: Any linear transformation between finite-dimensional spaces can be represented by a matrix relative to chosen bases.
• Invertibility: A linear transformation $T$ is invertible iff it is bijective, which is equivalent to having a non-zero determinant if represented by a square matrix.

💡 Key Takeaway

A linear transformation is a fundamental concept that preserves vector space operations; its properties—such as kernel, image, and invertibility—determine whether it establishes an isomorphism between spaces, revealing their structural equivalence.

📖 10. Dual Space & Dual Basis

🔑 Key Concepts & Definitions

• Dual Space (V*): The set of all linear functionals from a vector space V over a field F to F itself. Formally, $V^* = \{f: V \to F \mid f \text{ is linear}\}$.

• Linear Functional: A linear map $f: V \to F$, where V is a vector space over field F.

• Dual Basis: Given a basis $\{v_1, v_2, ..., v_n\}$ of V, the dual basis $\{f_1, f_2, ..., f_n\}$ in $V^*$ is defined by $f_i(v_j) = \delta_{ij}$ (Kronecker delta).

• Evaluation Map: For each $v \in V$, the map $\hat{v}: V^* \to F$ defined by $\hat{v}(f) = f(v)$. It links vectors and their dual functionals.

• Bidual Space (V**): The dual of the dual space $V^*$. There is a natural embedding $\iota: V \to V^{**}$ defined by $\iota(v)(f) = f(v)$.

📝 Essential Points

• The dual space $V^*$ is a vector space over the same field as V, with dimension equal to that of V if V is finite-dimensional.

• Dual Basis Construction: For a basis $\{v_1, ..., v_n\}$, the dual basis $\{f_1, ..., f_n\}$ satisfies $f_i(v_j) = \delta_{ij}$. Each $f_i$ is uniquely determined by this property.

• Isomorphism in Finite Dimensions: When V is finite-dimensional, $V$ and $V^*$ are isomorphic; the dual basis provides an explicit isomorphism.

• Bidual Isomorphism: For finite-dimensional V, the natural map $V \to V^{**}$ is an isomorphism, meaning V can be identified with its bidual.

• Dual Basis in Coordinates: If $v_j$ has coordinates $(a_{1j}, ..., a_{nj})$ in some basis, then the dual basis functionals $f_i$ can be represented as coordinate functionals extracting the ith coordinate.

💡 Key Takeaway

The dual space provides a powerful framework to analyze linear functionals and coordinate systems; in finite-dimensional spaces, the dual basis offers a concrete way to relate vectors and linear functionals, establishing an isomorphism between a space and its dual.

📖 11. Matrix Representation & Operations

🔑 Key Concepts & Definitions

• Matrix Representation: A way to express a linear transformation $T: V \to W$ as a matrix $A$ relative to chosen bases of $V$ and $W$. The matrix encodes how basis vectors are mapped under $T$.

• Matrix Multiplication: An operation combining two matrices $A$ and $B$ (of compatible sizes) to produce a new matrix $C = AB$, representing the composition of linear transformations.

• Transpose of a Matrix: The matrix obtained by swapping rows and columns of a matrix $A$, denoted $A^T$. It relates to dual spaces and adjoint operators.

• Rank of a Matrix: The maximum number of linearly independent rows or columns in a matrix $A$. It indicates the dimension of the image of the associated linear transformation.

• Determinant: A scalar value associated with a square matrix $A$, denoted $\det(A)$, indicating whether $A$ is invertible ($\det(A) \neq 0$) and relating to volume scaling.

• Inverse Matrix: For an invertible matrix $A$, the matrix $A^{-1}$ satisfying $AA^{-1} = I$, where $I$ is the identity matrix. Represents the inverse transformation.

📝 Essential Points

• The matrix representation depends on the choice of bases; different bases lead to different matrices for the same linear transformation.

• Matrix operations (addition, multiplication, transpose) correspond to algebraic operations on linear transformations: sum, composition, and adjoint.

• The rank of a matrix determines the dimension of the image of the linear transformation; full rank implies invertibility for square matrices.

• The determinant provides criteria for invertibility; a zero determinant indicates a singular matrix and a non-invertible transformation.

• Matrix multiplication models the composition of linear transformations: $(T_2 \circ T_1)$ corresponds to $A_{T_2} A_{T_1}$.

• The transpose relates to dual spaces and is used in defining adjoint operators, especially in inner product spaces.

• The inverse exists if and only if the matrix is square and non-singular ($\det(A) \neq 0$).

💡 Key Takeaway

Matrix representation provides a concrete algebraic framework to analyze and compute linear transformations, with operations like multiplication and transpose reflecting composition and duality. Understanding these operations is fundamental for solving systems, analyzing invertibility, and exploring properties like rank and determinants in linear algebra.

📖 12. Kernel, Image & Rank of Operators

🔑 Key Concepts & Definitions

• Kernel (Null Space): The set of all vectors $v$ in a vector space $V$ such that $T(v) = 0$, where $T: V \to W$ is a linear operator. Denoted as $\ker(T)$.

• Image (Range): The set of all vectors $w$ in $W$ for which there exists $v \in V$ with $T(v) = w$. Denoted as $\operatorname{Im}(T)$.

• Rank of an Operator: The dimension of the image of $T$, i.e., $\operatorname{rank}(T) = \dim(\operatorname{Im}(T))$.

• Nullity of an Operator: The dimension of the kernel of $T$, i.e., $\operatorname{nullity}(T) = \dim(\ker(T))$.

• Rank-Nullity Theorem: For a linear operator $T: V \to W$, $\dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)$.

📝 Essential Points

• The kernel measures the "loss" of information; vectors mapped to zero form a subspace called the null space.

• The image indicates the "reachable" vectors; its dimension (rank) reflects the operator's effectiveness.

• The rank-nullity theorem links the dimensions of the domain, kernel, and image, providing a fundamental relationship in linear algebra.

• For finite-dimensional spaces, the rank of $T$ is at most $\dim(V)$, and the nullity is at most $\dim(V)$.

• The properties of kernel and image are crucial in solving linear systems, understanding invertibility, and analyzing linear transformations.

• An operator is invertible iff its kernel is trivial ($\ker(T) = \{0\}$) and its rank equals $\dim(V)$.

💡 Key Takeaway

The kernel and image of a linear operator reveal its fundamental structure, with the rank-nullity theorem providing a vital link between these subspaces and the dimension of the domain, essential for understanding invertibility and the solution space of linear systems.

📊 Synthesis Tables

⚠️ Common Pitfalls & Confusions

1. Confusing the real and imaginary parts of a complex number with its modulus or argument.
2. Assuming all binary operations are commutative or associative without verification.
3. Overlooking the quadrant when calculating the argument of a complex number.
4. Mistaking the existence of an inverse for its uniqueness.
5. Confusing the properties of relations (reflexivity, symmetry, transitivity) and their implications.
6. Assuming the image of a linear transformation is always the entire codomain.
7. Forgetting that the identity element, if it exists, is unique.
8. Misapplying De Moivre's theorem to non-integer powers or roots.
9. Overgeneralizing properties of complex addition/multiplication to other algebraic systems.
10. Confusing the concepts of kernel, image, and rank in linear transformations.

✅ Exam Checklist

• Define and compute the real part, imaginary part, modulus, and argument of a complex number.
• Convert complex numbers between algebraic and polar forms.
• State and apply De Moivre's theorem for powers and roots.
• Describe properties of binary operations: associativity, commutativity, identity, and inverses.
• Determine whether a set with an operation forms a group, ring, or field.
• Characterize relations as reflexive, symmetric, transitive; identify equivalence relations.
• Define injective, surjective, and bijective functions; determine their images.
• Prove the uniqueness of the identity element in a set with a binary operation.
• Prove the uniqueness of inverses for elements in algebraic structures.
• Describe the concept of a linear subspace and verify subspace conditions.
• State the criteria for a subset to be a basis; relate to dimension.
• Compute the sum and direct sum of subspaces.
• Find the rank, kernel, and image of a linear transformation.
• Represent linear transformations with matrices; perform matrix operations.
• Determine if a linear transformation is an isomorphism.
• Find the dual space and dual basis of a vector space.
• Express linear transformations in matrix form and perform basis changes.