Study sheet: Group and Vector Space Algebra

📋 Course Outline

1. Matrix Definitions and Special Forms
2. Matrix Operations and Trace
3. Determinants and Their Properties
4. Invertible Matrices and Inverses
5. Linear Systems and Resolution
6. Internal Composition Laws
7. Groups and Group Morphisms
8. Subgroups and Their Characterizations
9. Group Morphisms and Isomorphisms
10. Vector Space Structures
11. Spanning Families, Independence, and Bases
12. Subspaces, Sums, and Rank
13. Linear Maps and Matrices
14. Eigenvalues and Diagonalization
15. Annihilating Polynomials
16. Diagonalization Criteria and Method
17. Triangularization Criteria and Method

📖 1. Matrix Definitions and Special Forms

🔑 Key Concepts & Definitions

• Matrix : A rectangular table with n rows and p columns whose entries belong to K, where K denotes R or C.
• Square matrix : A matrix with n rows and n columns.
• Identity matrix : The identity matrix In of order n is the diagonal matrix whose diagonal entries are all equal to 1.
• Symmetric matrix : A symmetric matrix is a square matrix M satisfying Mij = Mji for every pair of indices (i,j).

📖 2. Matrix Operations and Trace

🔑 Key Concepts & Definitions

• Matrix product : For A in Mnp(K) and B in Mpq(K), the product AB exists only when the number of columns of A equals the number of rows of B, and its entries satisfy $c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}$.
• Transpose : The transpose of A in Mnp(K) is the matrix tA in Mpn(K) whose rows, in order, are the columns of A.
• Trace : The trace of a square matrix A is the sum of its diagonal entries, $\operatorname{Tr}(A)=\sum_{i=1}^{n}a_{ii}$.

📝 Essential Points

📌 Matrix multiplication is associative and distributive, and the identity matrix satisfies AIn = A = InA.

💡 Memory Hook

Matrix multiplication is order-sensitive: generally AB ≠ BA, whereas addition is commutative.

📖 3. Determinants and Their Properties

🔑 Key Concepts & Definitions

• Minor : The determinant of the submatrix obtained by deleting row i and column j.

📝 Essential Points

📐 Formula — For a 2-by-2 matrix, \det\begin{pmatrix}a&b\c&d\end{pmatrix}=ad-bc.

• For a 3-by-3 determinant, expansion along a row or column expresses the determinant as signed entries multiplied by their corresponding minors.

📌 The determinant of a triangular matrix equals the product of its diagonal entries, and det(In)=1.

📌 Exchanging two rows or two columns multiplies a determinant by −1, while adding a linear combination of other rows or columns to one row or column leaves it unchanged.

💡 Memory Hook

Minor → cofactor sign → expansion → determinant.

📖 4. Invertible Matrices and Inverses

🔑 Key Concepts & Definitions

• Invertible matrix : A square matrix for which there exists a square matrix B satisfying AB = In = BA.
• Cofactor : The cofactor αij of the entry in position (i,j) is αij = (−1)i+jΔij, where Δij is its minor.

📝 Essential Points

📌 A square matrix A is invertible if and only if det(A) ≠ 0.

📐 Formula — For an invertible square matrix A, the inverse is $A^{-1}=\frac{1}{\det A}\,{}^{t}\!\operatorname{com}(A)$.

💡 Memory Hook

Nonzero determinant → invertibility → inverse by the adjugate formula.

📖 5. Linear Systems and Resolution

🔑 Key Concepts & Definitions

• Linear system : A system of n linear equations in p unknowns has coefficients aij, unknowns x1 through xp, and right-hand sides b1 through bn, all belonging to K.
• Rank : The rank of a matrix is the maximum size of a square submatrix with nonzero determinant, and the rank of a linear system is the rank of its coefficient matrix.

📝 Essential Points

📌 A linear system can be written in matrix form as AX = B, where A is the coefficient matrix, X is the unknown vector, and B is the right-hand-side vector.

📌 For a system of rank r with p unknowns, if r < p then p−r unknowns are treated as parameters, whereas if r ≥ p the unknowns are determined from p equations and checked against the remaining equations.

• Gaussian elimination solves a linear system by elementary row operations until the coefficient matrix becomes triangular, then solving from the simplest equations upward.

📐 Formula — For a Cramer system AX = B with det(A) ≠ 0, the solution satisfies $x_i=\frac{D_{x_i}}{\det A}$, where Dx_i replaces column i of A by B.

💡 Memory Hook

Matrix form → rank → Gaussian elimination or Cramer method.

📖 6. Internal Composition Laws

🔑 Key Concepts & Definitions

• Internal composition law : An application from E × E into E.
• Stable subset : A subset A of a set E with internal law ⋆ is stable if every pair x,y in A satisfies x⋆y ∈ A.
• Associativity : A law ⋆ on E is associative if a⋆(b⋆c) = (a⋆b)⋆c for every a,b,c in E.
• Identity element : An identity element e for ⋆ satisfies a⋆e = a = e⋆a for every a in E.
• Commutativity : A law ⋆ on E is commutative if x⋆y = y⋆x for every x,y in E.

💡 Memory Hook

Closure → associativity → identity → inverse → commutativity.

📖 7. Groups and Group Morphisms

🔑 Key Concepts & Definitions

• Group : A group is a nonempty set with an associative internal law possessing an identity element and an inverse for every element; if the law is also commutative, the group is abelian.
• Subgroup : A subgroup H of a group G is a subset that has a group structure under the law induced from G.
• Group morphism : A group morphism from (G,
• ) to (F,⋆) is a map f satisfying f(x
• y)=f(x)⋆f(y) for every x,y in G.
• Kernel : The kernel of a group morphism f from G to F is ker f = {x ∈ G : f(x) = eF}.
• Image : The image of a group morphism f from G to F is the set of values attained by f, Im f = {f(x) : x ∈ G}.

★ Must-know

📌 A nonempty subset H of a group (G,⋆) is a subgroup if and only if it is stable under ⋆, contains the identity, and contains the inverse of each of its elements.

Further detail

📌 A group morphism maps the identity of G to the identity of F and maps the inverse of x to the inverse of f(x).

💡 Memory Hook

A group requires inverses for every element; a subgroup retains the same operation inside a subset.

📖 8. Subgroups and Their Characterizations

🔑 Key Concepts & Definitions

• Subgroup : a nonempty subset of a group G that has a group structure under the law induced from G

★ Must-know

📌 A nonempty subset H of a group G is a subgroup if and only if it is closed under the group law, contains the identity element, and contains the inverse of every one of its elements.

📌 A subset H of a group G is a subgroup if and only if H is nonempty and, for every x and y in H, the element x⋆y⁻¹ belongs to H.

Further detail

• The inclusions (ℤ,+)⊂(ℚ,+)⊂(ℝ,+) and (ℚ*,×)⊂(ℝ*,×)⊂(ℂ*,×) give chains of subgroups.

💡 Memory Hook

Closure, identity, inverses

📖 9. Group Morphisms and Isomorphisms

🔑 Key Concepts & Definitions

• Group morphism : a map f:G→F such that f(x

• y)=f(x)⋆f(y) for all x,y∈G

• Kernel : the subset ker f={x∈G:f(x)=eF}

• Image : the subset Im f={f(x):x∈G} of F

★ Must-know

📌 A group morphism sends the identity of G to the identity of F and sends the inverse of x to the inverse of f(x).

📌 For a group morphism f:G→F, f is injective if and only if ker f={eG}, surjective if and only if Im f=F, and an isomorphism if and only if both conditions hold.

Further detail

• The composition of two group morphisms is again a group morphism.

💡 Memory Hook

Kernel controls injectivity; image controls surjectivity

📖 10. Vector Space Structures

🔑 Key Concepts & Definitions

• Vector space : a nonempty set E with an internal addition making (E,+) an abelian group and a scalar multiplication K×E→E satisfying distributivity, associativity of scalar multiplication, and 1

• u=u

★ Must-know

• The spaces ℝⁿ, Mₙₚ(ℝ), and ℂⁿ are vector spaces, with ℂⁿ being a vector space both over ℂ and, using real scalars, over ℝ.

Further detail

• The scalars K in the course are either ℝ or ℂ, and the elements of a K-vector space are called vectors.

• In ℝⁿ, addition and scalar multiplication are defined componentwise: (x₁,…,xₙ)+(y₁,…,yₙ)=(x₁+y₁,…,xₙ+yₙ) and λ(x₁,…,xₙ)=(λx₁,…,λxₙ).

📖 11. Spanning Families, Independence, and Bases

🔑 Key Concepts & Definitions

• Linear combination : A vector u is a linear combination of u₁,…,uₚ if there exist scalars λ₁,…,λₚ such that u=λ₁u₁+⋯+λₚuₚ.
• Spanning family : A family of vectors is spanning for E when every vector of E is a linear combination of the vectors in the family.
• Linearly independent family : A family (u₁,…,u_q) is linearly independent when λ₁u₁+⋯+λ_qu_q=0E implies λ₁=⋯=λ_q=0.
• Basis : an ordered family that is both linearly independent and spanning
• Dimension : The dimension of a vector space is the number of vectors in any one of its bases.

★ Must-know

📌 If B=(e₁,…,eₙ) is a basis, every vector u∈E has a unique representation u=α₁e₁+⋯+αₙeₙ.

Further detail

📌 A family is linearly dependent if and only if one of its vectors is a linear combination of the other vectors.

💡 Memory Hook

Spanning reaches every vector; independence prevents nontrivial zero combinations

📖 12. Subspaces, Sums, and Rank

🔑 Key Concepts & Definitions

• Vector subspace : a nonempty subset closed under vector addition and scalar multiplication
• Generated subspace : the set of all linear combinations of the vectors in A, denoted vect A
• Direct sum : Two subspaces E₁ and E₂ are supplementary when E₁+E₂=E and E₁∩E₂={0E}, written E=E₁⊕E₂.
• Rank of a family : The rank of a family A is the dimension of the subspace generated by A: rank A=dim(vect A).

📝 Essential Points

📌 A subset F of E is a vector subspace if and only if it is nonempty and αu+βv belongs to F for all u,v∈F and α,β∈K.

📐 Formula — For finite-dimensional subspaces E₁ and E₂, the dimension of their sum satisfies $\dim(E₁+E₂)=\dim E₁+\dim E₂-\dim(E₁\cap E₂)$.

💡 Memory Hook

Generated vectors produce a subspace, whose dimension gives the rank

📖 13. Linear Maps and Matrices

🔑 Key Concepts & Definitions

• Linear map : A map f:E→F between K-vector spaces is linear when f(u+v)=f(u)+f(v) and f(λu)=λf(u) for all u,v∈E and λ∈K.
• Matrix of a linear map : The matrix of f:E→F in bases B=(e₁,…,eₚ) and U=(u₁,…,uₙ) has as its columns the coordinate vectors of f(e₁),…,f(eₚ) in U.
• Kernel and image : For a linear map f:E→F, ker f={u∈E:f(u)=0F}, Im f={f(u):u∈E}, and rank f=dim(Im f).
• Isomorphism : a bijective linear map
• Change-of-basis matrix : The passage matrix P_{BB′} has as columns the coordinates in basis B of the vectors of the basis B′.

★ Must-know

📐 Formula — For composable linear maps f:E→F and g:F→G, their matrices satisfy $\operatorname{mat}_{B,V}(g\circ f)=\operatorname{mat}_{U,V}(g)\operatorname{mat}_{B,U}(f)$.

📐 Formula — For a linear map between finite-dimensional vector spaces, the rank theorem states $\dim E=\dim(\ker f)+\dim(\operatorname{Im} f)$.

📌 When E and F have the same finite dimension, a linear map is an isomorphism if and only if the determinant of its matrix in bases of E and F is nonzero.

Further detail

📌 A linear map sends the zero vector to the zero vector, and the composition of two linear maps is linear.

📐 Formula — For two bases B and B′, the passage matrices satisfy $P_{B′B}=P_{BB′}^{-1}$ and coordinate columns satisfy $X=P_{BB′}X′$.

💡 Memory Hook

Map → matrix → kernel and image → rank

📖 14. Eigenvalues and Diagonalization

🔑 Key Concepts & Definitions

• Eigenvalue : A scalar λ is an eigenvalue of an endomorphism f if there exists a nonzero vector u such that f(u)=λu.
• Eigenspace : The eigenspace associated with an eigenvalue λ is Eλ=ker(f−λIdE)={u∈E:f(u)=λu}.
• Characteristic polynomial : For a square matrix A of order n, the characteristic polynomial is P_A(λ)=det(A−λI_n).
• Diagonalizable matrix : A matrix is diagonalizable if it is similar to a diagonal matrix, and an endomorphism is diagonalizable if it has a basis in which its matrix is diagonal.

★ Must-know

📌 A scalar λ is an eigenvalue of A if and only if P_A(λ)=0, equivalently det(A−λI_n)=0.

📌 An endomorphism or matrix is diagonalizable if and only if its space has a basis formed by eigenvectors, and n distinct eigenvalues imply diagonalizability in dimension n.

Further detail

📌 The characteristic polynomial of a square matrix is an annihilating polynomial of that matrix by the Cayley–Hamilton theorem.

💡 Memory Hook

Characteristic roots produce eigenvectors, which can produce a diagonal form

📖 15. Annihilating Polynomials

🔑 Key Concepts & Definitions

• Annihilating polynomial : A polynomial P ∈ K[X] is an annihilating polynomial of A ∈ M_n(K) when $P(A)=0_n$.

📝 Essential Points

• For the matrix $A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, the polynomial $P(X)=X^3-X^2-X+I_2$ is annihilating because $A^2=I_2$ and $A^3=A$.

• For the given 3×3 matrix A, the characteristic determinant is $\det(A-\lambda I_3)=-(\lambda-4)^3$, so 4 is its only eigenvalue.

💡 Memory Hook

Characteristic polynomial vanishes on A → it annihilates A

📖 16. Diagonalization Criteria and Method

🔑 Key Concepts & Definitions

• Diagonalizable matrix : Similar to a diagonal matrix, while an endomorphism is diagonal in some basis of its vector space.

★ Must-know

• A matrix or endomorphism is diagonalizable if and only if its space has a basis consisting entirely of eigenvectors.

• If an n-dimensional matrix or endomorphism has n distinct eigenvalues, then it is diagonalizable.

• A matrix is diagonalizable if and only if its characteristic polynomial splits and each eigenspace dimension equals the algebraic multiplicity of its associated eigenvalue.

• To diagonalize an endomorphism, solve the characteristic equation, determine a basis of each eigenspace, form an eigenbasis, write the diagonal matrix in that basis, and state the similarity relation between the original and diagonal matrices.

Further detail

• For the matrix with characteristic polynomial $P_A(\lambda)=(5-\lambda)(1+\lambda)^2$, the eigenvalues are -1 and 5, with eigenspaces of dimensions 2 and 1 respectively, so the matrix is diagonalizable.

💡 Memory Hook

Eigenvalues → eigenspaces → eigenbasis → diagonal matrix

📖 17. Triangularization Criteria and Method

🔑 Key Concepts & Definitions

• Triangularizable matrix : Similar to a triangular matrix, while an endomorphism has a triangular matrix in some basis.

★ Must-know

📌 A matrix over K is triangularizable if and only if its characteristic polynomial splits over K, meaning that the characteristic equation has n roots counted with multiplicity.

• 🔄 The triangularization procedure is:
  1. Solve the characteristic equation.
  2. Determine bases of the eigenspaces.
  3. Choose a maximal linearly independent family of eigenvectors.
  4. Complete this family to a basis.
  5. Write the triangular matrix and state the similarity relation.

Further detail

• The matrix whose characteristic polynomial is $P_A(\lambda)=(2-\lambda)^3$ is triangularizable because its characteristic polynomial splits over the underlying field.

• For the linear map represented by the given 2×2 matrix in the canonical basis, the basis formed by u₁=(1,1) and u₂=(-1,1) yields the triangular matrix $T=\begin{pmatrix}2&0\\2&4\end{pmatrix}$.

💡 Memory Hook

Diagonalization requires an eigenbasis; triangularization only requires a triangular basis

📊 Synthesis Tables

Key matrix operations

Operation	Condition	Result
Addition	Same format	Entrywise sum
Scalar multiplication	Scalar in K	Each entry is multiplied by the scalar
Matrix product	Columns of A = rows of B	Entry cij is a row-column product
Transpose	Any matrix	Rows and columns are exchanged

Kernel, Image, and Isomorphism Criteria

Property	Criterion	Location
Injective	ker f={0}	Domain
Surjective	Im f=codomain	Codomain
Isomorphism	Injective and surjective	Both