CO1: Understand fundamental concepts of rings, subrings, integral domains, division rings, fields, ideals, and ring homomorphisms with their isomorphism theorems.
CO2: Master vector spaces, subspaces, linear span, linear independence, bases, dimension invariance, and quotient spaces over arbitrary fields.
CO3: Analyze linear transformations, null spaces, ranges, rank-nullity theorem, matrix representations, change of basis, characteristic equations, and the Cayley-Hamilton theorem.
CO4: Comprehend vector space isomorphisms, invertibility criteria, classification of finite-dimensional spaces, and the vector space isomorphism theorems.
📝Module:Unit 1 — Rings, Subrings, Integral Domains, Fields, and Ideals | 14
Axiomatic definition and elementary properties of rings, standard examples ($\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_n, R[x], M_n(R), C[a,b]$, Boolean rings), commutative rings, rings with unity, units and group of units $U(R)$. Subrings and subring criteria. Zero divisors, integral domains, cancellation laws, finite integral domains are fields, division rings (skew fields) and Hamilton's quaternions $\mathbb{H}$. Characteristic of a ring and characteristic of integral domains. Left, right, and two-sided ideals, ideal test, principal ideals. Algebra of ideals: sum, intersection, and product. Quotient rings (residue class rings) $R/I$ and coset operations.
📝Module:Unit 2 — Ring Homomorphisms and Isomorphism Theorems | 10
Ring homomorphisms: definition, types (monomorphisms, epimorphisms, isomorphisms, endomorphisms, automorphisms), and canonical examples (evaluation, reduction modulo $n$, canonical quotient projection $\pi$). Elementary properties of homomorphisms. Kernel and image of a homomorphism, kernel as a two-sided ideal, injectivity criterion ($\ker \phi = \{0\}$). First Isomorphism Theorem for rings ($R/\ker \phi \cong \operatorname{im} \phi$). Second Isomorphism Theorem ($(S+I)/I \cong S/(S \cap I)$). Third Isomorphism Theorem ($(R/I)/(J/I) \cong R/J$). Correspondence Theorem for ideals (lattice isomorphism of ideals).
Vector spaces over arbitrary fields: definition, axioms, elementary properties, and canonical examples ($F^n, P_n(F), F[x], M_{m \times n}(F), C[a,b]$). Subspaces and subspace criteria. Algebra of subspaces: intersection, union condition, and sum $W_1 + W_2$. Internal direct sum $W_1 \oplus W_2$ and uniqueness of representation. Linear combinations, linear span, and spanning sets. Linear dependence and independence. Bases of vector spaces, standard bases, minimal spanning and maximal independent characterizations. Steinitz Exchange Lemma, dimension invariance $\dim(V)$. Extension and reduction theorems. Grassmann's dimension formula: $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$. Quotient spaces $V/W$ and dimension theorem $\dim(V/W) = \dim(V) - \dim(W)$.
📝Module:Unit 4 — Linear Transformations, Rank-Nullity, and Eigenvalues | 14
Linear transformations: definition, preservation of origin and combinations, geometric examples (rotations, reflections, shears, projections) and differential/integral operators. Null space (kernel) and range (image), injectivity criterion. Rank-Nullity Theorem: $\operatorname{nullity}(T) + \operatorname{rank}(T) = \dim(V)$, algebraic complement decomposition. Algebra of transformations $\mathcal{L}(V, W)$ and composition. Matrix representation $[T]_{\mathcal{B}_V}^{\mathcal{B}_W}$, coordinate vectors, matrix operations, isomorphism $\mathcal{L}(V, W) \cong M_{m \times n}(F)$. Change of basis matrix and matrix similarity $[T]_{\mathcal{B}'} = P^{-1} [T]_{\mathcal{B}} P$. Eigenvalues, eigenvectors, and eigenspaces. Characteristic polynomial and characteristic equation $\det(A - \lambda I) = 0$, trace and determinant formulas. Linear independence of eigenvectors and diagonalizability. Cayley-Hamilton Theorem and applications to matrix powers and inverses.
📝Module:Unit 5 — Vector Space Isomorphisms and Invertibility | 8
Vector space isomorphisms: bijective linear maps, isomorphic classification ($V \cong W$ as equivalence relation). Invertibility criteria: left/right/two-sided inverses, linearity of $T^{-1}$, $\det([T]) \ne 0$. Classification of finite-dimensional vector spaces: $n$-dimensional space over $F$ is isomorphic to $F^n$, coordinate mapping $\phi_{\mathcal{B}}$. Invertibility vs. dimension theorems for finite-dimensional spaces (injectivity $\iff$ surjectivity $\iff$ invertibility) and infinite-dimensional shift operator counterexamples. First Isomorphism Theorem for vector spaces: $V/\ker(T) \cong \operatorname{im}(T)$ and Rank-Nullity recovery. Second Isomorphism Theorem: $(W_1 + W_2)/W_2 \cong W_1 / (W_1 \cap W_2)$ and Grassmann formula recovery. Third Isomorphism Theorem: $(V/W_1)/(W_2/W_1) \cong V/W_2$.
📝Textbooks
Gallian, Joseph A. (2021). Contemporary Abstract Algebra (10th ed.). Chapman and Hall/CRC.
Herstein, I. N. (1975). Topics in Algebra (2nd ed.). John Wiley & Sons.
Friedberg, Stephen H., Insel, Arnold J., & Spence, Lawrence E. (2019). Linear Algebra (5th ed.). Pearson.
Kumaresan, S. (2000). Linear Algebra: A Geometric Approach. Prentice-Hall of India.
Hoffman, Kenneth, & Kunze, Ray (1971). Linear Algebra (2nd ed.). Prentice-Hall.
Khanna, V. K., & Bhambri, S. K. (2017). A Course in Abstract Algebra (5th ed.). Vikas Publishing House.
02Rings, Subrings, Integral Domains, and IdealsCH
03Ring Homomorphisms and Isomorphism TheoremsCH
04Vector Spaces, Subspaces, Bases, and DimensionCH
05Linear Transformations, Rank-Nullity Theorem, and EigenvaluesCH