Linear Algebra: Vector Spaces

Linear Algebra

Vector Spaces

• Vector space defined over a field $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C}$) is a set $E$ with two operations: Addition : $E \times E \rightarrow E$, $(u, v) \mapsto u + v$ and Scalar multiplication : $\mathbb{K} \times E \rightarrow E$, $(\lambda, u) \mapsto \lambda u$.
• $(E, +)$ is an abelian group, which means associativity: $(u + v) + w = u + (v + w)$, identity element: $\exists 0_E \in E, \forall u \in E, u + 0_E = u$, inverse: $\forall u \in E, \exists -u \in E, u + (-u) = 0_E$ and commutativity: $u + v = v + u$.
• Scalar multiplication is distributive over vector addition: $\lambda(u + v) = \lambda u + \lambda v$
• Scalar multiplication is distributive over field addition: $(\lambda + \mu)u = \lambda u + \mu u$
• Scalar multiplication is associative: $\lambda(\mu u) = (\lambda \mu)u$
• Scalar multiplication has an identity element: $1_{\mathbb{K}}u = u$

Examples

• $\mathbb{K}^n$ is a vector space over $\mathbb{K}$
• The set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, denoted $C^0(\mathbb{R}, \mathbb{R})$, is a vector space over $\mathbb{R}$.
• The set of polynomials with coefficients in $\mathbb{K}$, denoted $\mathbb{K}[X]$, is a vector space over $\mathbb{K}$.

Vector Subspaces

• A subset $F$ of a vector space $E$ is a vector subspace of $E$ if $F$ is non-empty, is closed under addition: $\forall u, v \in F, u + v \in F$, and is closed under scalar multiplication: $\forall \lambda \in \mathbb{K}, \forall u \in F, \lambda u \in F$.
• $F$ is a vector subspace of $E$ if and only if $F$ is non-empty, and $\forall u, v \in F, \forall \lambda \in \mathbb{K}, \lambda u + v \in F$

Examples

• ${0_E}$ and $E$ are vector subspaces of $E$
• In $\mathbb{R}^2$, lines passing through the origin are vector subspaces

Linear Combinations and Spanning Sets

• A linear combination of vectors $u_1, \dots, u_n \in E$ is a vector of the form $\lambda_1 u_1 + \dots + \lambda_n u_n$, where $\lambda_1, \dots, \lambda_n \in \mathbb{K}$.
• The set of all linear combinations of $u_1, \dots, u_n$ is called the vector subspace spanned by $u_1, \dots, u_n$, denoted $\text{Vect}(u_1, \dots, u_n)$.
• A family of vectors $(u_i){i \in I}$ is said to span $E$ if $\text{Vect}((u_i){i \in I}) = E$.

Properties

• $\text{Vect}(u_1, \dots, u_n)$ is a vector subspace of $E$
• If $(u_i)_{i \in I}$ is a spanning set of $E$, then every vector in $E$ can be written as a linear combination of the vectors in the set.

Linear Independence and Bases

• A family of vectors $(u_1, \dots, u_n)$ is linearly independent if $\lambda_1 u_1 + \dots + \lambda_n u_n = 0_E$ implies $\lambda_1 = \dots = \lambda_n = 0$.
• A family of vectors is linearly dependent if it is not linearly independent.
• A basis of $E$ is a family of vectors that is both linearly independent and spanning.

Properties

• If $(u_1, \dots, u_n)$ is a basis of $E$, then every vector in $E$ can be written uniquely as a linear combination of $u_1, \dots, u_n$.
• If $E$ admits a finite basis, then all bases of $E$ have the same number of elements with this number being the dimension of $E$, represented by $\text{dim}(E)$.

Examples

• The canonical basis of $\mathbb{K}^n$ is $((1, 0, \dots, 0), (0, 1, \dots, 0), \dots, (0, 0, \dots, 1))$.
• The family $(1, X, X^2, \dots, X^n)$ is a basis of the vector space of polynomials of degree less than or equal to $n$, denoted $\mathbb{K}_n[X]$.

Linear Transformations

• For vector spaces $E$ and $F$ over $\mathbb{K}$, a map $f: E \rightarrow F$ is linear if $\forall u, v \in E, f(u + v) = f(u) + f(v)$ and $\forall \lambda \in \mathbb{K}, \forall u \in E, f(\lambda u) = \lambda f(u)$.

Properties

• $f(0_E) = 0_F$
• $f(\lambda u + v) = \lambda f(u) + f(v)$
• Set of linear maps from $E$ to $F$, denoted $\mathcal{L}(E, F)$, is a vector space over $\mathbb{K}$.

Kernel and Image

• Kernel of $f$ is $\text{Ker}(f) = {u \in E \mid f(u) = 0_F}$
• Image of $f$ is $\text{Im}(f) = {f(u) \mid u \in E}$.

Rank Theorem

• If $E$ is finite-dimensional, then $\text{dim}(E) = \text{dim}(\text{Ker}(f)) + \text{dim}(\text{Im}(f))$.

Matrices

• A matrix with $m$ rows and $n$ columns with coefficients in $\mathbb{K}$ is an array of the form: $$ A = \begin{pmatrix} a_{11} & \dots & a_{1n} \ \vdots & \ddots & \vdots \ a_{m1} & \dots & a_{mn} \end{pmatrix} $$
• The set of matrices with $m$ rows and $n$ columns with coefficients in $\mathbb{K}$ is denoted $\mathcal{M}_{m, n}(\mathbb{K})$.

Matrix Operations

• Addition: If $A, B \in \mathcal{M}{m, n}(\mathbb{K})$, then $A + B$ is the matrix whose coefficients are $(a{ij} + b_{ij})$.
• Scalar Multiplication: If $A \in \mathcal{M}{m, n}(\mathbb{K})$ and $\lambda \in \mathbb{K}$, then $\lambda A$ is the matrix whose coefficients are $(\lambda a{ij})$.
• Matrix Multiplication: If $A \in \mathcal{M}{m, n}(\mathbb{K})$ and $B \in \mathcal{M}{n, p}(\mathbb{K})$, then $AB$ is the matrix in $\mathcal{M}{m, p}(\mathbb{K})$ whose coefficients are $c{ij} = \sum_{k=1}^n a_{ik} b_{kj}$.

Matrices and Linear Transformations

• Any linear map $f: E \rightarrow F$ can be represented by a matrix in given bases of $E$ and $F$.
• Matrix multiplication corresponds to the composition of linear maps.

Determinant

• The determinant of a square matrix $A$ is a scalar characterizing the matrix's properties, such as invertibility. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, the determinant is $\text{det}(A) = ad - bc$.