Linear Algebra

• Vector Spaces:
  • Finite dimensional vector spaces over real or complex fields.
  • Subspaces.
  • Linear independence of vectors.
  • Span.
  • Basis, dimension.
  • Direct sum.
  • Dual space of a vector space.
• Linear Transformations:
  • Linear transformations and their matrix representations.
  • Range space, null space, rank, nullity.
  • Rank-nullity theorem.
  • Transpose of a linear transformation.
  • Adjoint of a linear transformation.
• Matrices:
  • Systems of linear equations.
  • Rank and inverse.
  • Determinant.
• Eigenvalues & Diagonalization:
  • Characteristic polynomial.
  • Eigenvalues and eigenvectors.
  • Diagonalization.
  • Minimal polynomial, Cayley-Hamilton Theorem.
  • Upper triangulation, triangular forms.
  • Nilpotent matrices.
  • Jordan canonical form.
• Inner Product Spaces:
  • Finite dimensional inner product spaces.
  • Gram-Schmidt orthonormalization process.
  • Scalar (dot) products, angle.
• Special Matrices & Forms:
  • Types of matrices.
  • Diagonalization by a unitary matrix.
  • Rotations.
  • GL_{n}, SL_{n}, O_{n}, SO_{2}, SO_{3}.
  • Bilinear and quadratic forms.Distributive properties

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Finite Dimensional Vector Spaces

Vector Space

A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold.

Commutativity

u + v = v + u for all u, v \in V.

Associativity

(u+v)+w=u+(v+w) and (ab)v=a(bv) for all u,v,w\in V and for all a,b\in F

Additive identity

There exists an element 0 \in V such that v + 0 = v for all v \in V

Additive inverse

For every v \in V, there exists w \in V such that v + w = 0

Multiplicative identity

1v = v for all v \in V

Distributive properties

a(u+v) = au + av and (a+b)v = av+bv for all a,b\in F and all u,v\in 𝑉

Subspace

Types of matrices

Matrix	Definition
Symmetric	a_{ij}=a_{ji}\implies A'=A
Skew Symmetric	a_{ij}=-a_{ji}\implies A'=-A
Triangular	all elements above or below the main diagonal are zero 1. Upper Triangular: elements below main diagonal are zero 2. Lower Triangular: elements above main diagonal are zero
Transpose (A',A^{T})	interchange of rows and columns
Orthogonal	AA'=I 1. If P,Q are orthogonal, P', P^{-1} and PQ are also orthogonal
Conjugate (\bar{A})	\bar{a}_{ij}= conjugate of a_{ij}
Conjugate Transpose (A^{\theta})	\bar{A}^{T}
Unitary	A^{\theta}A=I 1. If P is unitary, then \|P\| is of unit modulus and P',\bar{P},P^{\theta},P^{-1} are also unitary 2. Any two eigenvectors corresponding to distinct eigenvalues of a unitary matrix are orthogonal
Hermitian	a_{ij}= \bar{a}_{ji}
Skew Hermitian	a_{ij}= -\bar{a}_{ji}\implies A^{\theta}=-A 1. If A,B are hermitian, AB-BA is skew-hermitian 2. If A is any square matrix, A-A^* is skew-hermitian 3. Every square matrix can be uniquely represented as the sum of a hermitian and a skew-hermitian matrix 4. If A is a skew-hermitian (or hermitian) matrix, then iA is a hermitian matrix and \bar{A} is skew-hermitian (or hermitian)
Idempotent	A^2=A
Periodic	A^{k+1}=A where k\in\mathbb{Z}^+ 1. If k is the least positive integer for the condition, it is called the period of A. e.g. for idempotent matrix k=1
Involutory	A is its own inverse i.e. A^2=I
Nilpotent	A^k=0 where k\in\mathbb{Z}^+ 1. If is the least positive integer for the condition, it is called the index of A
Submatrix	matrix obtained from another matrix by delete rows and/or columns
Principal Diagonal	diagonal such that i=j
Bidiagonal	matrix with elements only on the main diagonal and either the diagonal above (super-diagonal) or the diagonal below (sub-diagonal)
Block	a matrix partitioned in sub-matrices
Block-diagonal	a block matrix with elements only on the diagonal
Boolean	all elements are 0 or 1
Permutation	a matrix representation of a permutation where every row and column contains only one element, namely, 1 and all other elements are 0
Vandermonde	each row contains elements which are terms of a geometric progression V^{m\times n}=\left[\begin{array}{cc}1 & \alpha_1 & \alpha_1^2 & \ldots & \alpha_{1}^{n-1} \\1 & \alpha_2 & \alpha_2^2 & \ldots & \alpha_{2}^{n-1} \\\vdots & \vdots & \vdots & \ldots & \vdots \\1 & \alpha_m & \alpha_m^2 & \ldots & \alpha_{m}^{n-1}\end{array}\right] determinant of a square Vandermonde matrix is det(V)=\prod_{1\leq i<j\leq n}(\alpha_j-\alpha_i)
Companion	companion matrix to a monic polynomial a(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}+x^n is C(a)=\left[\begin{array}{cc}0 & 0 & \ldots & 0 & -a_0 \\1 & 0 & \ldots & 0 & -a_1\\0 & 1 & \ldots & 0 & -a_2 \\\vdots & \vdots & \vdots & \ldots & \vdots\\0 & 0 & \ldots & 1 & -a_{n-1}\end{array}\right]