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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).