CO1: Appreciate the significance of factorization in rings and integral domains.
CO2: Compute the characteristic polynomial, eigenvalues, eigenvectors, and eigenspaces, as well as the geometric and the algebraic multiplicities of eigenvalues.
CO3: Compute inner products and determine orthogonality on vector spaces, including Gram-Schmidt orthogonalization to obtain orthonormal basis.
📝Module:Unit 1 — Polynomial Rings, Unique Factorization Domains, and Field of Quotients | 14
Field of quotients of an integral domain, construction and universal property, embedding of rings into quotient fields. Polynomial rings: formal power series, polynomial rings $R[x]$ over commutative ring $R$, degree and leading coefficient, polynomial rings over integral domains ($D[x]$ is an integral domain), units in $D[x]$. The Division Algorithm in $F[x]$ and consequences, The Remainder Theorem, The Factor Theorem, number of roots of a degree $n$ polynomial over a field bounded by $n$. Irreducible polynomials and reducible polynomials, associate elements, associates in $F[x]$. Primitive polynomials, content of a polynomial $c(f)$, Gauss's Lemma ($c(fg) = c(f)c(g)$ and irreducibility in $\mathbb{Z}[x] \iff \mathbb{Q}[x]$). Irreducibility tests: Eisenstein's Criterion, Modulo $p$ irreducibility test, Rational Roots Test, reducible polynomials of degree 2 and 3. Unique Factorization Domains (UFD): irreducible elements vs. prime elements in integral domains, Ascending Chain Condition on Principal Ideals (ACCP), $F[x]$ is a Principal Ideal Domain (PID) and hence a UFD, Gauss's Theorem ($R$ is a UFD implies $R[x]$ is a UFD), non-PID status of $\mathbb{Z}[x]$.
📝Module:Unit 2 — Dual Spaces, Linear Functionals, and Annihilators | 12
Linear functionals: definition $f: V \to F$, hyperplanes as kernels, coordinate and evaluation functionals, dual space $V^* = \mathcal{L}(V, F)$, dimension equality $\dim(V^*) = \dim(V)$ in finite dimensions. Dual basis: definition of dual basis $\mathcal{B}^* = \{f_1, \dots, f_n\}$ dual to basis $\mathcal{B} = \{v_1, \dots, v_n\}$ via $f_i(v_j) = \delta_{ij}$, coordinate expansion formulas $v = \sum_{i=1}^n f_i(v) v_i$ and $f = \sum_{j=1}^n f(v_j) f_j$. Double dual space $V^{**} = (V^*)^*$, canonical evaluation homomorphism $\psi: V \to V^{**}$ with $\psi(v)(f) = f(v)$, reflexivity and canonical isomorphism between $V$ and $V^{**}$ in finite dimensions. Annihilators: annihilator $S^0$ of subset $S \subseteq V$, subspace property of $S^0 \le V^*$, dimension theorem $\dim(W) + \dim(W^0) = \dim(V)$ for subspace $W$, sum and intersection relations ($(W_1 + W_2)^0 = W_1^0 \cap W_2^0$, $(W_1 \cap W_2)^0 = W_1^0 + W_2^0$), canonical isomorphisms $(V/W)^* \cong W^0$ and $W^* \cong V^*/W^0$. Transpose (dual map) of a linear transformation: definition $T^t: W^* \to V^*$ via $T^t(g) = g \circ T$, linearity, composition $(ST)^t = T^t S^t$, matrix representation of transpose in dual bases $[T^t]_{\mathcal{C}^*}^{\mathcal{B}^*} = ([T]_{\mathcal{B}}^{\mathcal{C}})^T$. Relations between fundamental subspaces: $\ker(T^t) = (\operatorname{im}(T))^0$, $\operatorname{im}(T^t) = (\ker(T))^0$, and equality of row rank and column rank ($\operatorname{rank}(T^t) = \operatorname{rank}(T)$).
Eigenvalues, eigenvectors, and eigenspaces of a linear operator $T: V \to V$: definition $T(v) = \lambda v$, eigenspace $E_\lambda = \ker(T - \lambda I)$, characteristic polynomial $p_T(\lambda) = \det(T - \lambda I)$, characteristic roots. Algebraic multiplicity $m_a(\lambda)$ and geometric multiplicity $m_g(\lambda) = \dim(E_\lambda)$, inequality $1 \le m_g(\lambda) \le m_a(\lambda)$. Linear independence of eigenvectors corresponding to distinct eigenvalues. Diagonalization of linear operators: necessary and sufficient criteria (characteristic polynomial splits and $m_g(\lambda) = m_a(\lambda)$ for each eigenvalue, or $V = \bigoplus_i E_{\lambda_i}$), diagonal matrix representation $[T]_\mathcal{B}$, change of basis matrix $P^{-1}AP = D$. Invariant subspaces ($T$-invariant subspaces): definition $T(W) \subseteq W$, restriction operator $T|_W$, block upper triangular matrix representations induced by invariant subspaces, direct sum decompositions into invariant subspaces. Cyclic subspaces $W = Z(v; T) = \operatorname{span}\{v, Tv, T^2v, \dots\}$, $T$-annihilator of a vector. The minimal polynomial: definition of monic polynomial $m_T(x)$ of least degree annihilating $T$, uniqueness, divisibility of any annihilating polynomial by $m_T(x)$, Cayley-Hamilton Theorem ($p_T(T) = 0 \implies m_T(x) \mid p_T(x)$), identical roots of $p_T(x)$ and $m_T(x)$. Minimal polynomial characterization of diagonalizability: $T$ is diagonalizable if and only if $m_T(x)$ factors into distinct linear factors $(x - \lambda_1)\dots(x - \lambda_k)$ over $F$.
Inner product spaces over $\mathbb{R}$ and $\mathbb{C}$: axiomatic definitions (conjugate symmetry / symmetry $\langle u, v \rangle = \overline{\langle v, u \rangle}$, linearity in first argument, positive-definiteness $\langle v, v \rangle \ge 0$ with equality iff $v = 0$), standard inner products on $\mathbb{R}^n$, $\mathbb{C}^n$, $C[a,b]$, and $M_n(F)$. Induced norm $\|v\| = \sqrt{\langle v, v \rangle}$ and metric $d(u,v) = \|u - v\|$, properties of norms. Cauchy-Schwarz Inequality: statement, complete algebraic proof over $\mathbb{R}$ and $\mathbb{C}$, and equality condition (linear dependence). Triangle Inequality $\|u + v\| \le \|u\| + \|v\|$. Parallelogram Law $\|u + v\|^2 + \|u - v\|^2 = 2\|u\|^2 + 2\|v\|^2$, Polarization Identities in real and complex inner product spaces. Orthogonality of vectors ($\langle u, v \rangle = 0$), Pythagorean Theorem for orthogonal sets. Orthogonal and orthonormal systems, linear independence of orthogonal systems of non-zero vectors. Orthonormal basis (ONB): definition, coordinate expressions relative to an ONB ($v = \sum_{i=1}^n \langle v, e_i \rangle e_i$, Parseval's identity for finite ONB). The Gram-Schmidt Orthogonalization Process: explicit constructive algorithm converting a linearly independent set $\{v_1, \dots, v_k\}$ into an orthonormal set $\{e_1, \dots, e_k\}$, normalization, QR-decomposition of matrices. Orthogonal complements: definition $W^\perp = \{v \in V : \langle v, w \rangle = 0\ \forall w \in W\}$, subspace property of $W^\perp$, closedness, orthogonal decomposition theorem $V = W \oplus W^\perp$ for finite-dimensional subspaces $W$, $(W^\perp)^\perp = W$. Orthogonal projection $P_W: V \to W$, projection formula, best approximation theorem ($\|v - P_W(v)\| \le \|v - w\|$ for all $w \in W$). Bessel's Inequality for finite and arbitrary orthonormal sequences $\sum_{i} |\langle v, e_i \rangle|^2 \le \|v\|^2$.
📝Module:Unit 5 — Adjoint Operators, Least Squares, and Spectral Theory | 14
The adjoint of a linear operator: Riesz Representation Theorem for linear functionals on finite-dimensional inner product spaces ($\exists! y \in V$ such that $f(x) = \langle x, y \rangle$), definition and existence of the unique adjoint operator $T^*: V \to V$ satisfying $\langle T(u), v \rangle = \langle u, T^*(v) \rangle$, algebraic properties ($(S+T)^* = S^* + T^*$, $(cT)^* = \bar{c}T^*$, $(ST)^* = T^*S^*$, $(T^*)^* = T$, $I^* = I$, $(T^{-1})^* = (T^*)^{-1}$). Matrix representation of adjoint in an orthonormal basis: $[T^*]_\mathcal{B} = ([T]_\mathcal{B})^*$ (conjugate transpose $A^* = \bar{A}^T$). Relations between fundamental subspaces: $\ker(T^*) = (\operatorname{im}(T))^\perp$, $\operatorname{im}(T^*) = (\ker(T))^\perp$. Least squares approximation: inconsistent linear systems $Ax = b$, normal equations $A^*Ax = A^*b$, existence of least squares solutions, projection onto the column space $\operatorname{Col}(A)$, geometric interpretation. Minimal solutions to systems of linear equations: underdetermined consistent systems, unique minimal norm solution $x_0 \in \operatorname{Row}(A) = (\ker(A))^\perp$, pseudo-inverse (Moore-Penrose inverse) connection. Normal operators: definition $TT^* = T^*T$, properties ($\|T(v)\| = \|T^*(v)\|$, $T(v) = \lambda v \iff T^*(v) = \bar{\lambda} v$, eigenspaces for distinct eigenvalues are mutually orthogonal). Self-adjoint (Hermitian) operators: definition $T^* = T$, real eigenvalues, existence of orthonormal basis of eigenvectors for self-adjoint operators over $\mathbb{R}$ and $\mathbb{C}$. Unitary and orthogonal operators: definition $T^*T = TT^* = I$, isometry property ($\|T(v)\| = \|v\|$ and $\langle T(u), T(v) \rangle = \langle u, v \rangle$), eigenvalues lie on unit circle ($|\lambda| = 1$), orthogonal matrices ($A^TA = I$) and unitary matrices ($U^*U = I$). Spectral Theorem: Complex Spectral Theorem (a linear operator on a finite-dimensional complex inner product space is normal if and only if there exists an orthonormal basis of eigenvectors for $V$), Real Spectral Theorem (a linear operator on a finite-dimensional real inner product space is self-adjoint if and only if there exists an orthonormal basis of eigenvectors for $V$). Spectral decomposition $T = \sum_{i=1}^k \lambda_i P_i$. Positive operators, orthogonal projections, and introduction to polar and singular value decompositions.
📝Textbooks
Gallian, Joseph A. (2019). Contemporary Abstract Algebra (9th ed.). Cengage Learning India Private Limited.
Friedberg, Stephen H., Insel, Arnold J., & Spence, Lawrence E. (2022). Linear Algebra (5th ed.). Pearson Education.
Herstein, I. N. (2006). Topics in Algebra (2nd ed.). Wiley Student Edition. India.
Hoffman, Kenneth M. & Kunze, Ray Alden (1978). Linear Algebra (2nd ed.). Prentice-Hall of India Pvt. Limited. Delhi. Pearson Education India Reprint, 2015.
Kumaresan, S. (2000). Linear Algebra: A Geometric Approach. Prentice Hall India Learning Private Limited, New title Edition.
Khanna, V. K. & Bhambri, S. K. (2017). A Course in Abstract Algebra (5th ed.). Vikas Publishing House.
Strang, Gilbert (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press.
02Polynomial Rings, Factorization, and Integral DomainsCH
03Linear Functionals, Dual Spaces, and AnnihilatorsCH
04Eigenspaces, Diagonalization, and Minimal PolynomialsCH
05Inner Product Spaces and OrthogonalityCH
06Adjoint Operators, Least Squares, and Spectral TheoryCH