Understanding Vector Spaces and Their Axioms

Vector Spaces

• Defined as a set $V$ with addition ($+: V \times V \rightarrow V$) and scalar multiplication ($\cdot: \mathbb{R} \times V \rightarrow V$) operations.
• Addition and scalar multiplication operations must satisfy specific axioms like commutativity, associativity, existence of a zero vector and additive inverse, and distributivity.

Axioms for Vector Spaces:

• Commutativity: For all $u, v \in V$, $u + v = v + u$.
• Associativity: For all $u, v, w \in V$, $(u + v) + w = u + (v + w)$.
• Zero Vector: There exists $0 \in V$ such that for all $v \in V$, $v + 0 = v$.
• Additive Inverse: For all $v \in V$, there exists $w \in V$ such that $v + w = 0$.
• Associativity (Scalar Multiplication): For all $a, b \in \mathbb{R}$, for all $v \in V$, $(a \cdot b) \cdot v = a \cdot (b \cdot v)$.
• Multiplicative Identity: For all $v \in V$, $1 \cdot v = v$.
• Distributivity (Scalar over Vector Addition): For all $a \in \mathbb{R}$, for all $u, v \in V$, $a \cdot (u + v) = a \cdot u + a \cdot v$.
• Distributivity (Scalar Addition): For all $a, b \in \mathbb{R}$, for all $v \in V$, $(a + b) \cdot v = a \cdot v + b \cdot v$.

Examples of Vector Spaces:

• $\mathbb{R}^n = {(x_1, \dots, x_n) \mid x_i \in \mathbb{R}}$
• $\mathbb{C}^n = {(z_1, \dots, z_n) \mid z_i \in \mathbb{C}}$
• $M_{m \times n}(\mathbb{R})$: real $m \times n$ matrices.
• $P_n(\mathbb{R})$: polynomials of degree $\leq n$ with real coefficients.
• $F(\mathbb{R}, \mathbb{R})$: functions from $\mathbb{R}$ to $\mathbb{R}$.

Counterexamples of Vector Spaces:

• $\mathbb{R}^2$ with $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1y_2)$ is not a vector space.
• $\mathbb{R}^2$ with $a \cdot (x, y) = (ax, y)$ is not a vector space.

Subspaces

• A subset $W$ of a vector space $V$.
• Must satisfy these conditions: $0 \in W$, $u + v \in W$ for all $u, v \in W$, and $a \cdot v \in W$ for all $a \in \mathbb{R}$ and $v \in W$.
• If $W$ is a subspace of $V$, $W$ is also a vector space with the same operations as $V$.

Examples of Subspaces:

• $W = {(x, 0) \mid x \in \mathbb{R}} \subset \mathbb{R}^2$ is a subspace.
• $W = {(x, 1) \mid x \in \mathbb{R}} \subset \mathbb{R}^2$ is not a subspace because $(0, 0) \notin W$.
• $W = {(x, x^2) \mid x \in \mathbb{R}} \subset \mathbb{R}^2$ is not a subspace because the sum of elements in $W$ is not necessarily in $W$.
• $W = {f \in C(\mathbb{R}, \mathbb{R}) \mid f(3) = 0} \subset C(\mathbb{R}, \mathbb{R})$ is a subspace, given $C(\mathbb{R}, \mathbb{R})$ is the set of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$.
• $W = {f \in C(\mathbb{R}, \mathbb{R}) \mid f(3) = 1} \subset C(\mathbb{R}, \mathbb{R})$ is not a subspace because the zero function is not in $W$.

Linear Combinations

• Given $v_1, \dots, v_n \in V$, $a_1 v_1 + \dots + a_n v_n$ is a linear combination, where $a_1, \dots, a_n \in \mathbb{R}$.
• The span of $v_1, \dots, v_n$ is the set of all possible linear combinations of $v_1, \dots, v_n$.
• Denoted as $\text{span}(v_1, \dots, v_n) = {a_1 v_1 + \dots + a_n v_n \mid a_1, \dots, a_n \in \mathbb{R}}$.
• $\text{span}(v_1, \dots, v_n)$ is a subspace of $V$.

Linear Independence

• Vectors $v_1, \dots, v_n \in V$ are linearly independent if the equation $a_1 v_1 + \dots + a_n v_n = 0$ only has the trivial solution $a_1 = a_2 = \dots = a_n = 0$.
• If vectors are not linearly independent, they are linearly dependent.
• There exist $a_1, \dots, a_n \in \mathbb{R}$, not all zero, such that $a_1 v_1 + \dots + a_n v_n = 0$ for linearly dependent vectors.