Comprehensive Notes on Linear Algebra and Affine Geometry

DIAGONALIZATION OF ENDOMORPHISMS

• Fundamental Definitions:
  • Let $K$ be a field and $V$ a $K$-vector space of finite dimension $n = \dim_K(V)$.
  • An endomorphism is a linear map $f: V \rightarrow V$.
  • The set of all such maps is denoted $\text{End}_K(V)$.
• Coordinate Matrix and Base Change:
  • Let $B$ and $B'$ be bases of $V$. Let $A$ be the matrix of $f$ in base $B$ and $A'$ be the matrix of $f$ in base $B'$.
  • For a vector $v \in V$, let $X$ be its coordinates in $B$ and $Y$ be the coordinates of its image $f(v)$.
  • The coordinate expression is $Y = AX$.
  • If $P$ is the change of base matrix ($X_B = P X_{B'}$), then substituting yields $P Y' = A P X'$.
  • This implies $Y' = (P^{-1} A P) X'$, proving that the relation between coordinate matrices is $A' = P^{-1} A P$.
• Similar Matrices:
  • Two matrices $A, B \in M_n(K)$ are similar (denoted $A \sim B$) if there exists an invertible matrix $P \in GL_n(K)$ such that $B = P^{-1} A P$.
  • Similarity is an equivalence relation:
    • Reflexive: $A = I^{-1} A I$.
    • Symmetric: If $B = P^{-1} A P$, then $A = (P^{-1})^{-1} B P^{-1}$.
    • Transitive: If $B = P^{-1} A P$ and $C = Q^{-1} B Q$, then $C = (PQ)^{-1} A (PQ)$.
  • Similar matrices have the same rank: $rg(A) = rg(B)$.

EIGENVALUES AND EIGENVECTORS

• Diagonalizability:
  • An endomorphism $f$ is diagonalizable if there exists a base $B$ of $V$ such that its coordinate matrix is diagonal ($D = \text{diag}(\lambda_1, \dots, \lambda_n)$).
  • In this base, the image of the basis vectors follows the rule $f(e_i) = \lambda_i e_i$.
  • A matrix $A$ is diagonalizable if it is similar to a diagonal matrix.
• Definitions of Eigen-elements:
  • An element $\lambda \in K$ is an eigenvalue of $f$ if there exists a non-zero vector $0 \neq v \in V$ such that $f(v) = \lambda v$.
  • The vector $v$ is called an eigenvector associated with the eigenvalue $\lambda$.
• Characterization of Diagonalizability (TFAE):
  1. $f$ is diagonalizable.
  2. Every coordinate matrix of $f$ is diagonalizable.
  3. There exists a base formed by eigenvectors of $f$.
• Eigenspaces (Subespacios Propios):
  • For an eigenvalue $\lambda$, the eigenspace is $S(\lambda) = \{ v \in V \mid f(v) = \lambda v \} = \text{ker}(\lambda \text{id}_V - f)$.
  • The dimension of the eigenspace is the geometric multiplicity $d_\lambda = \dim S(\lambda) = n - rg(\lambda I_n - A)$.
  • An eigenvalue must satisfy the equation $|\lambda I - A| = 0$.

THEOREMS ON DIAGONALIZATION

• The Multiplicity Inequality:
  • Let $a_\lambda$ be the multiplicity of $\lambda$ as a root of the characteristic polynomial (algebraic multiplicity).
  • Fundamental Lemma: $d_\lambda \leq a_\lambda$.
  • Proof Sketch: By extending a base of $S(\lambda)$ to a base of $V$, the matrix of $f$ takes a block form, showing $(x - \lambda)^{d_\lambda}$ divides the characteristic polynomial.
• Main Diagonalization Theorem:
  • Let $\lambda_1, \dots, \lambda_r$ be distinct eigenvalues. $f$ is diagonalizable if and only if:
    1. All roots of the characteristic polynomial belong to $K$.
    2. $a_{\lambda_i} = d_{\lambda_i}$ for all $1 \leq i \leq r$.
    3. The sum of dimensions $\sum_{i=1}^r d_{\lambda_i} = n$.
• Applications and Properties:
  • If $f$ has $n$ distinct eigenvalues, it is automatically diagonalizable.
  • Similar matrices share the same characteristic polynomial.
  • Calculation of matrix powers: If $A = P D P^{-1}$, then $A^k = P D^k P^{-1}$.
• The Fibonacci Example:
  • Defined by $a_1 = 1$, $a_2 = 1$, $a_{n+1} = a_n + a_{n-1}$.
  • Vector recurrence: $\begin{pmatrix} a_{n+1} \\ a_n \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} a_n \\ a_{n-1} \end{pmatrix}$.
  • General term: $\begin{pmatrix} a_{n+2} \\ a_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} a_2 \\ a_1 \end{pmatrix}$.

BILINEAR FORMS

• Definition: A bilinear form $F: V \times V \rightarrow K$ is an application such that:
  1. $F(v, \alpha w + \beta u) = \alpha F(v, w) + \beta F(v, u)$.
  2. $F(\alpha v + \beta w, u) = \alpha F(v, u) + \beta F(w, u)$.
• Coordinate Matrix of a Bilinear Form:
  • For base $B = \{e_1, \dots, e_n\}$, matrix $A$ has entries $a_{ij} = F(e_i, e_j)$.
  • The calculation follows: $F(v, w) = X^t A Y$, where $X, Y$ are coordinate vectors of $v, w$.
• Congruence of Matrices:
  • Under basis change $X_B = P X_{B'}$, the matrix transforms as $A' = P^t A P$.
  • Matrices $A$ and $B$ are congruent if there exists $P \in GL_n(K)$ such that $B = P^t A P$.
  • Congruence is an equivalence relation and preserves rank: $rg(F)$.

SYMMETRIC BILINEAR FORMS

• Symmetry: $F$ is symmetric if $F(u, v) = F(v, u)$ for all $u, v \in V$.
  • This corresponds to the matrix being symmetric ($A = A^t$).
• Orthogonality:
  • Vectors $u, v$ are orthogonal if $F(u, v) = 0$.
  • $u$ is isotropic if $F(u, u) = 0$.
  • Orthogonal subspace of $S$: $S^\perp = \{v \in V \mid F(v, s) = 0 \, \forall s \in S\}$.
  • Characterization: $v \in S^\perp \iff F(v, b) = 0$ for all elements $b$ in a base of $S$.
• Regular Subspaces:
  • $S$ is regular if $S \cap S^\perp = 0$.
  • If $S$ is regular, then $V = S \oplus S^\perp$.
• Diagonalization of Bilinear Forms:
  • If $\text{char}(K) \neq 2$, every symmetric bilinear form admits an orthogonal base (where the matrix is diagonal).
  • Proof by induction on dimension: Uses the existence of a non-isotropic vector $b$ to split the space into $\langle b \rangle \oplus \langle b \rangle^\perp$.

SYLVESTER'S LAW OF INERTIA

• Standard Forms in $\mathbb{R}$:
  • For $F \in \text{BilSim}_\mathbb{R}(V)$, there exists a base where the matrix is $D = \text{diag}(1, \dots, 1, -1, \dots, -1, 0, \dots, 0)$.
  • The number of $1$s ($r$) and $-1$s ($s$) are unique for a given form.
  • Signature: denoted as $sig(F) = r$ (or sometimes the triple $(r, s, n-r-s)$).
• Sylvester's Theorem: Matrices are congruent if and only if they have the same rank and signature.
• Scalar Product: A symmetric bilinear form on $\mathbb{R}$ is a scalar product if $rg(F) = sig(F) = n$.
  • This implies it is positive definite: $F(u, u) > 0$ for all $u \neq 0$.

EUCLIDEAN SPACES

• Euclidean Space: A real vector space provided with a scalar product $\cdot$.
• Norm of a Vector: $\lVert v \rVert = \sqrt{v \cdot v}$.
  • Properties: $\lVert v \rVert \geq 0$, $\lVert v \rVert = 0 \iff v = 0$, $\lVert \lambda v \rVert = |\lambda| \lVert v \rVert$.
• Fundamental Inequalities:
  • Cauchy-Schwarz: $|v \cdot w| \leq \lVert v \rVert \lVert w \rVert$.
  • Triangle Inequality: $\lVert v + w \rVert \leq \lVert v \rVert + \lVert w \rVert$.
• Angles: The angle $\alpha$ between $v, w$ satisfies $\cos(\alpha) = \frac{v \cdot w}{\lVert v \rVert \lVert w \rVert}$.
• Pythagorean Theorem: If $v \cdot w = 0$, then $\lVert v + w \rVert^2 = \lVert v \rVert^2 + \lVert w \rVert^2$.

ORTHOGONALIZATION AND BASES

• Gram-Schmidt Process: Technique to transform any base $B = \{e_1, \dots, e_n\}$ into an orthogonal base $B' = \{\bar{e}_1, \dots, \bar{e}_n\}$.
  1. $\bar{e}_1 = e_1$.
  2. $\bar{e}_k = e_k - \sum_{j=1}^{k-1} \frac{e_k \cdot \bar{e}_j}{\lVert \bar{e}_j \rVert^2} \bar{e}_j$.
• Orthonormal Base: An orthogonal base where every vector is unary ($\lVert v \rVert = 1$).
  • Changing between orthonormal bases involves an orthogonal matrix $A$ where $A^{-1} = A^t$.
  • The set of such matrices is the orthogonal group $O_n(\mathbb{R})$.
• Orthogonal Projection: For a subspace $U$, the projection of $v$ is the unique vector $p_U(v) \in U$ such that $v - p_U(v) \in U^\perp$.
  • Best Approximation: $p_U(v)$ is the vector in $U$ closest to $v$. $\lVert v - p_U(v) \rVert \leq \lVert v - w \rVert$ for all $w \in U$.

ISOMETRIES

• Definition: A linear map $h$ is an isometry if it preserves the scalar product: $h(v) \cdot h(w) = v \cdot w$.
• Equivalent Definitions:
  1. $h$ is an isometry.
  2. $h$ preserves norms: $\lVert h(v) \rVert = \lVert v \rVert$.
  3. The coordinate matrix $A$ in an orthonormal base is orthogonal ($A^t A = I$).
• Isometries in $\mathbb{R}^2$:
  • Case 1 (Determinant -1): Symmetries. Diagonalizable with eigenvalues $1, -1$.
  • Case 2 (Determinant 1): Rotations. Matrix $A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$. Represents a rotation of angle $\theta$.

AFFINE SPACES

• Definition: A triple $(A, V, +)$ where $A$ is a non-empty set of points and $V$ is a vector space.
  • Axiom 1: For any $P, Q \in A$, there exists a unique vector $u = \vec{PQ} \in V$ such that $Q = P + u$.
  • Axiom 2: $P + (u + w) = (P + u) + w$.
• Chasles Relation: $\vec{PQ} + \vec{QR} = \vec{PR}$.
• Reference Systems: A pair $(O, B)$ where $O$ is an origin point and $B$ is a base for $V$.
  • Coordinates of point $P$ are simply coordinates of vector $\vec{OP}$.
• Coordinate Change in Affine Systems:
  • Expression: $X_R = Z_R + A X_{R'}$ where $A$ is base change and $Z_R$ are coordinates of original origin in new system.
  • Compact Matrix Form: $\begin{pmatrix} 1 \\ X_R \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ Z & A \end{pmatrix} \begin{pmatrix} 1 \\ X_{R'} \end{pmatrix}$.

AFFINE VARIETIES

• Definition: A variety centered at $P$ with direction subspace $W$ is $W_P = \{P + w \mid w \in W \}$.
  • Dim 0: Point; Dim 1: Line; Dim 2: Plane; Dim $n-1$: Hyperplane.
• Representations:
  1. Vectorial: $Q = P + \lambda_1 u_1 + \dots + \lambda_r u_r$.
  2. Parametric: System of equations in coordinates with parameters $\lambda_i$.
  3. Implicit (Cartesian): Linear system of equations setting constraints on coordinates.
• Intersection: If non-empty, $W_P \cap U_Q = (W \cap U)_R$ for any $R$ in the intersection.
• Affine Span: The smallest variety containing a set of points $\mathcal{P} = \{P_1, \dots, P_t\}$.
  • Points are affinely independent if vectors $\vec{P_1 P_2}, \dots, \vec{P_1 P_t}$ are linearly independent.

AFFINITIES AND MOVEMENTS

• Affinities (Affine Maps):
  • A map $f: A_1 \rightarrow A_2$ is an affinity if there is an underlying linear map $\bar{f}: V_1 \rightarrow V_2$ such that $f(Q) = f(P) + \bar{f}(\vec{PQ})$.
  • They preserve parallel structures and varieties.
• Fixed Points: $P$ is fixed if $f(P) = P$. The set of fixed points forms an affine variety $Fix(f) = (\text{ker}(\bar{f} - Id))_P$.
• Matrix of an Affinity: $Y = B + AX$. Compactly: $\begin{pmatrix} 1 \\ Y \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ B & A \end{pmatrix} \begin{pmatrix} 1 \\ X \end{pmatrix}$.
• Group Structure: Affinities form a group under composition ($GA(A)$).
• Special Families:
  • Translations: $t_v(P) = P + v$. Underlying map is Identity.
  • Homotheties: $h_{C, \alpha}(P) = C + \alpha \vec{CP}$. Underlying map is $\alpha Id$.
• Euclidean Affine Spaces:
  • Distance: $d(P, Q) = \lVert \vec{PQ} \rVert$.
  • Distance between point and variety: $d(P, W_Q) = \lVert \vec{P p_{W_Q}(P)} \rVert$.
  • Distance between varieties: $d(W_P, U_Q) = d(P, (W + U)_Q)$.
• Movements: An affinity that is also a distance isometry. This occurs if and only if the underlying linear map $\bar{f}$ is an isometry ($\bar{f} \in O(V)$).