Linear Algebra: Vector Spaces and Applications

• The text covers fundamental concepts in linear algebra and analytic geometry.

Basic Definitions

• An vector space over a field $\mathbb{K}$ is a set $E$ equipped with addition and scalar multiplication operations.
• Addition is from $E \times E$ to $E$, represented by $(u, v) \mapsto u + v$.
• Scalar multiplication is from $\mathbb{K} \times E$ to $E$, represented by $(\lambda, u) \mapsto \lambda u$.
• The field $\mathbb{K}$ is often $\mathbb{R}$ or $\mathbb{C}$.
• Eight axioms must be satisfied for vector spaces: Associativity and commutativity of addition, existence of an additive identity (zero vector) and additive inverse, distributivity of scalar multiplication with respect to vector and scalar addition, associativity of scalar multiplication, and existence of a multiplicative identity (1).

Applications

• A linear application $f$ between vector spaces $E$ and $F$ over the same field $\mathbb{K}$ satisfies $f(u + v) = f(u) + f(v)$ and $f(\lambda u) = \lambda f(u)$ for all $u, v \in E$ and $\lambda \in \mathbb{K}$.

Matrices

• A matrix is a rectangular array of numbers.
• An $m \times n$ matrix $A$ has $m$ rows and $n$ columns, with elements denoted as $a_{ij}$.

Key Concepts

• A base of a vector space $E$ is a set of linearly independent vectors that span $E$.
• The dimension of $E$ is the number of vectors in a base of $E$.
• For a square matrix $A$, an eigenvector $v$ satisfies $Av = \lambda v$ for a scalar eigenvalue $\lambda$.
• The dot product of vectors $u$ and $v$ is denoted $\langle u, v \rangle$ and fulfills defined positive, symmetry, and bilinearity.
• Two vectors are orthogonal if their dot product is zero.

Important Theorems

• The dimension theorem states that for a linear map $f : E \rightarrow F$, $\dim(E) = \dim(\text{Ker}(f)) + \dim(\text{Im}(f))$, where $\text{Ker}(f)$ is the kernel and $\text{Im}(f)$ is the image of $f$.
• The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial.
• A matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$.