Chapter 3: Vector Spaces and Linear Mappings (Part I)

Definition and Fundamental Axioms of Vector Spaces

• General Definition: A non-empty set $E(+, \cdot)$ is defined as a vector space over a field $K$ if it satisfies specific axioms related to internal and external operations. The elements of the field $K$ are referred to as scalars, while the elements of the set $E$ are called vectors.

• Internal Law Analysis (Addition): The operation "+" represents the internal law. For $E$ to be a vector space, $(E, +)$ must be an abelian group, meaning it must satisfy the following five axioms:     * Closure: For all $u, v \in E$, the sum $u + v$ must also be in $E$. This signifies that $E$ is stable or closed under addition.     * Associativity: For all $u, v, w \in E$, the grouping of elements does not change the result: $(u + v) + w = u + (v + w)$.     * Identity Element: There exists an element $0_E \in E$ such that for all $u \in E$, $u + 0_E = 0_E + u = u$. By default, any vector space must contain at least this element. This identity element is unique and is called the zero vector of $E$.     * Inverse or Symmetric Element: For every $u \in E$, there exists a unique element $v \in E$ (written as $v = -u$ or $u = -v$) such that $u + v = v + u = 0_E$.     * Commutativity: For all $u, v \in E$, the order of addition does not matter: $u + v = v + u$. If all five axioms are met, $(E, +)$ is an abelian or commutative group.

• External Law Analysis (Scalar Multiplication): The operation "." represents the external law. The following axioms must be satisfied:     * Closure under Scalar Multiplication: For all $u \in E$ and all scalars $\lambda \in K$, the product $\lambda u$ must be in $E$.     * Compatibility with Multiplication in $K$: For all $u \in E$ and for all $\lambda, \gamma \in K$, results are consistent: $(\lambda \cdot \gamma) \cdot u = \lambda \cdot (\gamma \cdot u)$.     * Multiplicative Identity: For all $u \in E$, multiplying by the scalar $1$ yields the same vector: $1 \times u = u \times 1 = u$.

• Axioms for Mixed Operations: These axioms relate both internal (addition) and external (multiplication) laws:     * Distributivity over Addition of Scalars: For all $u \in E$ and all $\lambda, \gamma \in K$, the operation follows $(\lambda + \gamma) \cdot u = \lambda \cdot u + \gamma \cdot u$.     * Distributivity over Addition of Vectors: For all $u, v \in E$ and all $\lambda \in K$, the operation follows $\lambda \cdot (u + v) = \lambda \cdot u + \lambda \cdot v$.

The Vector Space $\mathbb{R}^n$

• Definition of $\mathbb{R}^n$: Let $n \in \mathbb{N}$ be a natural number where $n \ge 1$. The set $\mathbb{R}^n$ consists of all $n$-tuples $x$, also called vectors, represented in the form $x = (x_1, x_2, \dots, x_n)$. In this form, each $x_i \in \mathbb{R}$ is called the $i$-th component of the vector.

• Standard Operations in $\mathbb{R}^n$:     * Addition of Vectors: For all $x, y \in \mathbb{R}^n$ where $x = (x_1, x_2, \dots, x_n)$ and $y = (y_1, y_2, \dots, y_n)$, addition is defined component-wise: $x + y = (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n)$.     * Multiplication by a Scalar: For all $x \in \mathbb{R}^n$ and all scalars $\lambda \in \mathbb{R}$, multiplication is defined component-wise: $\lambda x = (\lambda x_1, \lambda x_2, \dots, \lambda x_n)$.

• Properties of $\mathbb{R}^n$: The set $\mathbb{R}^n$ is a vector space over the field of real numbers (an $\mathbb{R}$-vector space) because it satisfies:     * $(x + y) + z = x + (y + z)$ (Associativity).     * $x + y = y + x$ (Commutativity).     * $x + 0 = 0 + x$ (Zero vector identity).     * $x + (-x) = -x + x = 0$ (Additive inverse).     * $\lambda(\gamma x) = \lambda \gamma x$ (Compatibility).     * $(\lambda + \gamma) x = \lambda x + \gamma x$ (Scalar distributivity).     * $\lambda(x + y) = \lambda x + \lambda y$ (Vector distributivity).     * $1 \cdot x = x \cdot 1 = x$ (Scalar unit identity).

Vector Subspaces and Linear Combinations

• Definition of a Subspace: Let $F$ be a non-empty subset of an $\mathbb{R}$-vector space $E$. $F$ is called a vector subspace of $E$ if for all $u, v \in F$ and for all $\alpha, \beta \in \mathbb{R}$, the linear combination $\alpha u + \beta v$ also belongs to $F$.

• Requirement for Subspaces: If $F$ is a subspace of $E$, then $F$ is itself an $\mathbb{R}$-vector space.

• Alternative Criteria for Subspaces: A non-empty subset $F$ is a subspace of $E$ if and only if:     1. For all $u, v \in F$, the sum $u + v \in F$.     2. For all $u \in F$ and $\alpha \in \mathbb{R}$, the product $\alpha u \in F$.     * This is equivalent to the condition: $u + \alpha v \in F$ and $\alpha u + v \in F$.

• Trivial and Basic Subspaces:     * The set containing only the zero vector, ${0_E}$, is always a subspace of $E$.     * The intersection of two subspaces $F$ and $G$ ($F \cap G$) is always a subspace of $E$.     * The union of two subspaces $F \cup G$ is not necessarily a subspace. It is a subspace if and only if one is contained within the other (either $F \subseteq G$ or $G \subseteq F$).

• The Concept of Span: For vectors $x_1, x_2, \dots, x_n \in E$, the set defined as $\text{Span}{x_1, x_2, \dots, x_n} = {\alpha_1 x_1 + \alpha_2 x_2 + \dots + \alpha_n x_n \mid \alpha_i \in \mathbb{R}}$ is the smallest vector space containing these vectors.     * A vector $x$ belongs to the span if and only if there exist scalars such that $x = \sum_{i=1}^{n} \alpha_i x_i$.     * Geometric Interpretation: If $u \in E$ and $u \neq 0_E$, $\text{Span}{u} = {\alpha u \mid \alpha \in \mathbb{R}}$. This represents a straight line passing through the origin, collinear with vector $u$.

Sums and Direct Sums of Subspaces

• Sum of Two Subspaces: Let $F$ and $G$ be subspaces of $E$. Their sum is defined as $F + G = {x \mid \exists u \in F, \exists v \in G \text{ such that } x = u + v}$. The sum $F + G$ is a subspace of $E$ and is the smallest vector space containing both $F$ and $G$.

• Direct Sum Definition: Subspaces $F$ and $G$ are in a direct sum, denoted $F \oplus G$, if every vector $u \in F + G$ can be uniquely decomposed as $u = f + g$, where $f \in F$ and $g \in G$.     * Uniqueness Condition: If $u = f + g$ and $u = f' + g'$ (where $f, f' \in F$ and $g, g' \in G$), then it must follow that $f = f'$ and $g = g'$.

• Property of Direct Sums: Subspaces $F$ and $G$ are in a direct sum if and only if their intersection contains only the zero vector: $F \cap G = {0_E}$.

• Supplementary Subspaces: Subspaces $F$ and $G$ are called supplementary in $E$ if every vector in the entire space $E$ can be uniquely decomposed as a sum of a vector from $F$ and a vector from $G$. This is equivalent to saying $E = F \oplus G$, which requires:     1. $F + G = E$     2. $F \cap G = {0_E}$

Linear Independence and Generating Sets

• Linearly Independent Set: A set of vectors ${u_1, u_2, \dots, u_k} \subset E$ is linearly independent if the equation $\alpha_1 u_1 + \alpha_2 u_2 + \dots + \alpha_k u_k = 0$ implies that all scalars must be zero ($\alpha_1 = 0, \alpha_2 = 0, \dots, \alpha_k = 0$).

• Linearly Dependent Set: A set is linearly dependent if there exist scalars $(\alpha_1, \dots, \alpha_k)$, not all of which are zero, such that $\sum_{i=1}^k \alpha_i u_i = 0$.

• Specific Independence Examples:     * Single Vector: A set containing one vector ${u}$ is independent iff $u \neq 0_E$. It is dependent iff $u = 0_E$.     * Two Vectors: A set containing two vectors ${u, v}$ is independent iff they are not collinear. In other words, they are dependent iff one is a scalar multiple of the other ($\exists \alpha \in \mathbb{R}^* / u = \alpha v$).

• Generating Set: A set ${u_1, u_2, \dots, u_k}$ is a generating set of $E$ if every vector $u \in E$ can be expressed as a linear combination of these vectors ($E = \text{Span}{u_1, \dots, u_k}$).

Basis and Dimension in Finite Spaces

• Finite Dimensional Vector Space: A vector space that possesses a finite generating set.

• Basis of a Vector Space: A set $B = {e_1, e_2, \dots, e_n}$ is a basis for $E$ if it satisfies two conditions:     1. It is linearly independent.     2. It is a generating set of $E$.

• Coordinates: If $B$ is a basis, then every $u \in E$ has a unique set of coordinates $(\alpha_1, \alpha_2, \dots, \alpha_n) \in \mathbb{R}^n$ such that $u = \sum_{i=1}^n \alpha_i e_i$.

• Dimension Definition: The dimension of a finite dimensional vector space is the number of vectors in any basis set. The cardinality (number of elements) is identical for all bases of a given space.     * The dimension of the trivial space ${0_E}$ is $0$.

• Fundamental Dimensional Theorems (Theorem 4): In a space $E$ with $\dim(E) = n$:     * Any independent set contains at most $n$ vectors.     * Any independent set of exactly $n$ vectors is automatically a basis.     * Any generating set contains at least $n$ vectors.     * Any generating set of exactly $n$ vectors is automatically a basis.     * Every generating set contains a basis (can be reduced by removing vectors).     * Every independent set can be extended to form a basis (by adding vectors).

Canonical Bases and Rank

• Canonical Basis of $\mathbb{R}^n$: Defined by vectors $e_i = (0, 0, \dots, 1, 0, \dots, 0)$ where the $1$ is at the $i$-th position. Any vector $x = (x_1, \dots, x_n)$ can be written as $x = \sum x_i e_i$. The dimension $\dim(\mathbb{R}^n) = n$.

• Space of Matrices $M_{m, n}(\mathbb{R})$: The set of all $m \times n$ real matrices is a vector space with $\dim(M_{m, n}(\mathbb{R})) = m \times n$.     * Canonical Basis of Matrices: Consists of matrices $E_{ij}$ that have a $1$ at position $(i, j)$ and zeroes elsewhere.

• Subspaces in Finite Spaces (Theorem 5): If $E$ is finite dimensional, any subspace $F$ is also finite dimensional, and $0 \le \dim(F) \le \dim(E)$. If $\dim(F) = \dim(E)$, then $F = E$.

• Rank Definition: For a subset $F = {x_1, x_2, \dots, x_k}$, the rank of $F$ is the dimension of the subspace generated by $F$, denoted $\text{rank}(F) = \dim(\text{Span}{x_1, \dots, x_k})$.

• Rank Properties (Theorem 6):     * $\text{rank}(F) \le k$ (number of vectors).     * $\text{rank}(F) \le n$ (dimension of the space).     * $\text{rank}(F) = p$ implies independent subset properties; specifically, $\text{rank}(F) = k$ iff $F$ is independent.     * $\text{rank}(F) = n$ iff $F$ is a generating set of $E$.

• Grassman's Identity and Supplementary Condition:     * Theorem 8: $\dim(F + G) = \dim(F) + \dim(G) - \dim(F \cap G)$.     * Theorem 7: $F$ and $G$ are supplementary iff $\dim(F) + \dim(G) = \dim(E)$ AND $\dim(F \cap G) = 0$.

Matrix Representation of Vectors and Rank

• Coordinate Representation: For a basis $B = {e_1, e_2, \dots, e_n}$, a vector $u$ is represented by the column vector $\text{Mat}_B(u) = (\alpha_1, \alpha_2, \dots, \alpha_n)^T$.

• Matrix of a Set of Vectors: For a set $F = {u_1, u_2, \dots, u_p}$, the matrix $\text{Mat}_B(F)$ is an $n \times p$ matrix where each column is the coordinate vector of $u_k$ in basis $B$.

• Rank of a Matrix: Defined as the rank of the set of vectors formed by its columns. A square matrix of size $n$ is invertible if and only if $\text{rank}(A) = n$.

• Gaussian Elimination and Row Echelon Form:     * Row Echelon Form Criteria:         1. All-zero rows are at the bottom.         2. The leading entry (pivot) of a non-zero row is to the right of the pivot of the row above it.     * Rank Calculation: The rank of a matrix in row echelon form equals the number of non-zero rows. For example, a matrix with 3 non-zero rows in echelon form has a rank of 3.     * Elementary Row Operations (used to reach echelon form without changing rank):         1. Permutation of any two rows.         2. Multiplication of a row by a non-zero scalar.         3. Adding a linear combination of other rows to a row.

Change of Basis

• Transformation Matrix Definition: Let $B$ be the old basis and $B'$ be the new basis. The matrix $P_B^{B'}$ is formed by taking the coordinates of the vectors of the new basis $B'$ expressed in terms of the old basis $B$.

• Properties of Transformation Matrices:     * $P_B^{B'}$ is always invertible.     * The inverse matrix $P_{B'}^B = (P_B^{B'})^{-1}$ transforms coordinates from the new basis back to the old basis.

• Change of Basis Formula: Let $X$ be the coordinates in basis $B$ and $X'$ be the coordinates in basis $B'$. The relationship is: $X = P_B^{B'} X'$ and $X' = P_{B'}^B X$.

• Example in $\mathbb{R}^3$: If $B$ is the canonical basis and new vectors are $e_1' = e_1 + e_2$, $e_2' = e_1 - e_2$, and $e_3' = e_3$, the matrix is:     $P_B^{B'} = \begin{pmatrix} 1 & 1 & 0 \ 1 & -1 & 0 \ 0 & 0 & 1 \end{pmatrix}$

• Example in $\mathbb{R}^2$: Given $B = {e_1, e_2}$ and $B' = {e_1 + e_2, e_1 - e_2}$. To find coordinates of $x$ in $B'$, the relationship is determined by inverting the matrix, leading to:     $\begin{pmatrix} x_1' \ x_2' \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix}$