I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract Algebra. Here are the course descriptions for each class: Linear Algebra Chapter 1 Vector Spaces 1.1 Introduction 1.2 Vector Spaces 1.3 Subspaces 1.4 Linear Combinations and Systems of Linear Equations 1.5 Linear Dependence and Linear Independence 1.6 Bases and Dimension Chapter 2 Transformations and Matrices 2.1 Linear Transformations, Null Spaces, and Ranges 2.2 The Matrix Representation of a Linear Transformation 2.3 Composition of Linear Transformations and Matrix Multiplication 2.4 Invertibility and Isomorphisms 2.5 The Change of Coordinates Matrix Chapter 3 Elementary Matrix Operations and Systems of Linear Equations 3.1 Elementary Matrix Operations and Elementary Matrices 3.2 The Rank of a Matrix and Matrix Inverses 3.3 Systems of Linear Equations – Theoretical Aspects 3.4 Systems of Linear Equations – Computational Aspects Index of Definitions Chapter 4 Determinants 4.4 Summary – Important Facts about Determinants Chapter 5 Diagonalization 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalizability Chapter 6 Inner Product Spaces 6.1 Inner Products and Norms 6.2 The Gram-Schmidt Orthogonalization Process Chapter 7 Canonical Forms 7.1 The Jordan Canonical Form I 7.2 The Jordan Canonical Form II 7.3 The Minimal Polynomial Abstract Algebra I A. Group Theory Binary Operations Groups Subgroups Permutation Groups Orbits and Cycles Cyclic Groups Cosets and Lagrange Homomorphisms Isomorphisms and Cayley’s Theorem Factor Groups Fundamental Homomorphism Theorem B. Rings Rings and Fields Integral Domains Little Fermat and Euler Theorems Fields of Quotients Polynomial Rings Polynomial and Division Algorithm Remainder Theorem/Factor Theorem Homomorphisms and Factor Rings Prime and Maximal Ideals and PIDs, Prime Ideals C. Field Theory (Introduction) Field Extensions Existence of a Splitting Field Constructibility with Ruler and Compass –Trisecting Angles I'm currently taking a course called "Matrix Algebra", which covers things from Linear Algebra such as Matrices, determinants, vector spaces, eigenvalues, orthogonality, but it's mainly computations based while the Linear Algebra course above is more proof based, proving theorems then computations. But are the proofs similar to Abstract Algebra and how hard would a proof based Linear Algebra course be compared to the first part of Abstract Algebra? Only curious because many people say Abstract Algebra is more difficult than any Algebra course because it's very proof oriented while a Linear Algebra course is more applied (there are proofs in my Matrix Algebra course, but my professor is not going to make us prove anything on a test, just applications and calculations).