Vector Spaces and Linear Algebra Review

Vocabulary practice flashcards covering fundamental definitions, theorems, and concepts of vector spaces, linear maps, bases, dimensions, direct sums, and hyperplanes.

$\mathbb{K}$-Vector Space ($\mathbb{K}$-espace vectoriel)

A triplet $(E, +, \cdot)$ where $+$ is an internal law on $E$ such that $(E, +)$ is an abelian group, and $\cdot$ is an external law defined from $\mathbb{K} \times E$ to $E$ satisfying distributivity over vector and scalar addition, associativity of scalar multiplication, and $1 \cdot x = x$ for all $x \in E$.

Vector Subspace (Sous-espace vectoriel)

A non-empty subset $F$ of a vector space $E$ containing $0_E$ and stable under linear combinations, satisfying $\forall (x, y) \in F^2, \forall \lambda \in \mathbb{K}, x + \lambda y \in F$.

Linear Map / Morphism of Vector Spaces (Application linéaire)

A mapping $f$ from a vector space $E$ to a vector space $F$ that preserves addition and scalar multiplication, satisfying $\forall (x, y) \in E^2, \forall \lambda \in \mathbb{K}, f(x + \lambda y) = f(x) + \lambda f(y)$.

$$\mathcal{L}(E, F)$$

The set of all linear maps from a $\mathbb{K}$-vector space $E$ into a $\mathbb{K}$-vector space $F$, which forms a vector subspace of $F^E$.

Kernel of a Linear Map ($\ker(f)$)

The vector subspace of $E$ comprising all vectors mapped to $0_F$ by $f$, defined as $\ker(f) = f^{-1}(\{0_F\}) = \{x \in E \mid f(x) = 0_F\}$.

Image of a Linear Map ($\operatorname{Im}(f)$)

The vector subspace of $F$ formed by all outputs attained by $f$, defined as $\operatorname{Im}(f) = f(E) = \{y \in F \mid \exists x \in E, f(x) = y\}$.

Linearly Independent Family / Free Family (Famille libre) - Finite

A finite family $(e_1, \dots, e_n)$ of vectors in $E$ such that $\forall (\lambda_1, \dots, \lambda_n) \in \mathbb{K}^n$, $\sum_{i=1}^n \lambda_i e_i = 0_E \implies \lambda_1 = \dots = \lambda_n = 0$.

Sub-family (Sous-famille)

A family $(\varepsilon_j)_{j \in J}$ of vectors in $E$ is a sub-family of $(e_i)_{i \in I}$ if and only if there exists an injection $\varphi: J \to I$ such that $\forall j \in J, \varepsilon_j = e_{\varphi(j)}$.

Almost Null Family (Famille presque nulle)

A family $(\lambda_i)_{i \in I}$ of scalars in $\mathbb{K}$ for which the set of non-zero elements $\{i \in I \mid \lambda_i \neq 0\}$ is finite, with the set of all such families denoted $\mathbb{K}^{(I)}$.

Linearly Independent Family (Famille libre) - General Case

A family $(e_i)_{i \in I}$ of vectors in $E$ such that for every almost null family of scalars $(\lambda_i)_{i \in I} \in \mathbb{K}^{(I)}$, $\sum_{i \in I} \lambda_i e_i = 0 \implies \forall i \in I, \lambda_i = 0$.

Linear Span / Subspace Spanned by a Subset ($\operatorname{Vect}(A)$)

The smallest vector subspace (with respect to inclusion) of $E$ containing the subset $A$, consisting of all linear combinations of elements of $A$.

Spanning Family / Generating Family (Famille génératrice)

A family $(e_1, \dots, e_n)$ of vectors in $E$ whose linear span equals a vector subspace $F$, meaning $\operatorname{Vect}(e_1, \dots, e_n) = F$.

Super-family (Sur-famille)

A family $\mathcal{F}$ of vectors is a super-family of $\mathcal{F}'$ when $\mathcal{F}'$ is a sub-family of $\mathcal{F}$; any super-family of a spanning family of $E$ is also spanning for $E$.

Basis (Base)

A family of vectors in a vector space $E$ (or subspace $F$) that is simultaneously linearly independent (free) and spanning (generating).

Family of Coordinates (Famille des coordonnées)

The unique family of scalars $(\lambda_i)_{i \in I}$ such that a vector $x \in E$ decomposes as $x = \sum_{i \in I} \lambda_i e_i$ relative to a given basis $\mathcal{B} = (e_i)_{i \in I}$.

Finite-Dimensional Vector Space (Espace de dimension finie)

A vector space $E$ that admits a finite spanning (generating) family.

Dimension of a Vector Space ($\dim(E)$)

The common number of vectors contained in any basis of a finite-dimensional vector space $E$.

Rank of a Family of Vectors ($\operatorname{rg}(e_1, \dots, e_n)$)

The dimension of the vector subspace spanned by the family, defined as $\operatorname{rg}(e_1, \dots, e_n) = \dim(\operatorname{Vect}(e_1, \dots, e_n))$.

Rank of a Linear Map ($\operatorname{rg}(f)$)

The dimension of the image of a linear map $f$, defined as $\operatorname{rg}(f) = \dim(\operatorname{Im}(f))$ when $\operatorname{Im}(f)$ is finite-dimensional.

Isomorphic Vector Spaces (Espaces isomorphes)

Two vector spaces $E$ and $F$ for which there exists an isomorphism (a bijective linear map) from $E$ to $F$; if $E$ is finite-dimensional, this is equivalent to $F$ being finite-dimensional and $\dim(E) = \dim(F)$.

Sum of Subspaces ($\sum_{i=1}^n F_i$)

The vector subspace of $E$ consisting of all elements $x$ that can be written in the form $x = \sum_{i=1}^n x_i$ with $(x_1, \dots, x_n) \in \prod_{i=1}^n F_i$.

Direct Sum of Subspaces ($\bigoplus_{i=1}^n F_i$)

A sum of subspaces for which the map $(x_1, \dots, x_n) \mapsto \sum_{i=1}^n x_i$ is injective, equivalently stating that $\sum_{i=1}^n x_i = 0_E \implies x_1 = \dots = x_n = 0_E$.

Supplementary Subspaces (Sous-espaces supplémentaires)

Two subspaces $F$ and $G$ of $E$ satisfying $F \oplus G = E$, meaning that every $x \in E$ can be written uniquely as $x = x_F + x_G$ with $x_F \in F$ and $x_G \in G$.

Grassmann's Formula (Formule de Grassmann)

The formula relating the dimensions of two finite-dimensional subspaces $F$ and $G$: $\dim(F + G) = \dim(F) + \dim(G) - \dim(F \cap G)$.

Rank Theorem (Théorème du rang)

The fundamental result stating that for $f \in \mathcal{L}(E, F)$ with $E$ finite-dimensional, $\dim(E) = \dim(\ker(f)) + \dim(\operatorname{Im}(f))$.

Vector Projection (Projection vectorielle)

Given $E = F \oplus G$, the linear map $p: E \to E$ defined by $p(x) = x_F$, characterized by $p \circ p = p$, $\ker(p) = G$, and $\operatorname{Im}(p) = F$.

Vector Symmetry / Reflection (Symétrie vectorielle)

Given $E = F \oplus G$, the linear map $s: E \to E$ defined by $s(x) = x_F - x_G$, characterized by being an involution ($s \circ s = \operatorname{Id}$) where $\operatorname{Inv}(s) = F$ and $\operatorname{Opp}(s) = G$.

Hyperplane (Hyperplan)

A vector subspace $H$ of $E$ for which there exists a vector line $D$ such that $H \oplus D = E$; in an $n$-dimensional space, this is equivalent to $\dim(H) = n - 1$.

Linear Form (Forme linéaire)

Any linear mapping from a $\mathbb{K}$-vector space $E$ into its scalar field $\mathbb{K}$.

Dual Space ($E^*$)

The space $\mathcal{L}(E, \mathbb{K})$ of all linear forms on $E$, which satisfies $\dim(E^*) = \dim(E)$ when $E$ is finite-dimensional.