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MTH212Sciences3 Unitsintermediate

Linear Algebra

This course introduces students to the fundamental concepts of linear algebra. It covers vector spaces, linear transformations, matrices, determinants, eigenvalues, and eigenvectors. Students will learn how to perform matrix operations, solve systems of linear equations, and analyze vector spaces. The course aims to provide a solid foundation for further studies in mathematics, engineering, and physics.

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91h
Study Time
13
Weeks
7h
Per Week
intermediate
Math Level
Course Keywords
Linear AlgebraVector SpacesMatricesEigenvaluesDeterminants

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
📊
Math Level
Moderate Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Vector Spaces

2

Linear Transformations

3

Matrices

4

Determinants

5

Eigenvalues

6

Eigenvectors

Total Topics6 topics

Requirements

Knowledge and skills recommended for success

Basic Algebra

Calculus I

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (2 methods)

Computer-Based Tests

Comprehensive evaluation of course material understanding

Written Assessment

Final Examination

Comprehensive evaluation of course material understanding

Computer Based Test

Career Opportunities

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Data Analyst

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Software Engineer

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Financial Analyst

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Operations Research Analyst

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Statistician

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Industry Applications

Real-world sectors where you can apply your knowledge

Computer ScienceEngineeringFinanceData SciencePhysics

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Vector Spaces

4h

Unit 1: Vector Spaces

4 study hours
  • •Read the introduction to vector spaces.
  • •Understand the definitions and examples of vector spaces.
  • •Study spaces associated with vector spaces.
  • •Define and understand vector subspaces.
Week
2

Module 1: Vector Spaces

4h

Unit 2: Linear Combinations

4 study hours
  • •Define linear combinations of column vectors.
  • •Form linear combinations using different scalars.
  • •Understand the consistency of a system.
  • •Define and understand the linear span of a collection of vectors.
Week
3

Module 1: Vector Spaces

5h

Unit 3: Linear Transformation I

5 study hours
  • •Verify the linearity of mappings between vector spaces.
  • •Construct linear transformations with specified properties.
  • •Define the range and kernel of a linear transformation.
  • •Calculate the rank and nullity of a linear operator.
Week
4

Module 1: Vector Spaces

5h

Unit 4: Linear Transformation II

5 study hours
  • •Prove and apply the Rank Nullity Theorem.
  • •Define an isomorphism between two vector spaces.
  • •Show that two vector spaces are isomorphic if and only if they have the same dimension.
  • •Prove and use the fundamental theorem of homomorphism.
Week
5

Module 2: Matrices

5h

Unit 1: Matrices I

5 study hours
  • •Define and give examples of various types of matrices.
  • •Obtain a matrix associated with a given linear transformation.
  • •Define a linear transformation given its associated matrix.
  • •Evaluate the sum, difference, product, and scalar multiples of matrices.
Week
6

Module 2: Matrices

5h

Unit 2: Matrices II

5 study hours
  • •Obtain the transpose and conjugate of a matrix.
  • •Determine if a given matrix is invertible.
  • •Obtain the inverse of a matrix.
  • •Discuss the effect of change of basis on the matrix of a linear transformation.
Week
7

Module 2: Matrices

5h

Unit 3: Matrices III

5 study hours
  • •State the invertible matrix theorem.
  • •State and prove the conditions for a matrix to be invertible.
  • •Define and obtain the rank of a matrix.
  • •Reduce a matrix to echelon form.
Week
8

Module 3: Determinants

4h

Unit 1: Determinants I

4 study hours
  • •Define the determinant of a matrix.
  • •Evaluate the determinant of a square matrix using properties of determinants.
  • •Obtain the minor, cofactors, and adjoint of a square matrix.
Week
9

Module 3: Determinants

4h

Unit 2: Determinants II

4 study hours
  • •Compute the inverse of an invertible matrix using its adjoint.
  • •Apply Cramer's rule to solve systems of linear equations.
  • •Understand the product formula for determinants.
  • •Define and evaluate the determinant rank of a matrix.
Week
10

Module 4: Eigenvalues and Eigenvectors

5h

Unit 1: Eigenvalues and Eigenvectors

5 study hours
  • •Obtain the characteristic polynomial of a linear transformation or a matrix.
  • •Obtain the eigenvalues, eigenvectors, and eigenspaces of a linear transformation or a matrix.
Week
11

Module 4: Eigenvalues and Eigenvectors

5h

Unit 1: Eigenvalues and Eigenvectors

5 study hours
  • •Obtain a basis of a vector space with respect to which the matrix of a linear transformation is in diagonal form.
  • •Obtain a non-singular matrix P which diagonalizes a given diagonalizable matrix A.
Week
12

Module 4: Eigenvalues and Eigenvectors

5h

Unit 2: Characteristic and Minimal Polynomials

5 study hours
  • •State and prove the Cayley-Hamilton theorem.
  • •Find the inverse of an invertible matrix using the Cayley-Hamilton theorem.
  • •Prove that a scalar is an eigenvalue if and only if it is a root of the minimal polynomial.
Week
13

Module 4: Eigenvalues and Eigenvectors

5h

Unit 2: Characteristic and Minimal Polynomials

5 study hours
  • •Obtain the minimal polynomial of a matrix (or linear transformation) if the characteristic polynomial is known.
  • •Review all modules and units.
  • •Work on assignments and exercises.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Review definitions of vector spaces, linear transformations, and matrices.

2

Practice solving systems of linear equations using Gaussian elimination and Cramer's rule.

3

Focus on understanding the properties of determinants and their applications.

4

Master the computation of eigenvalues and eigenvectors for various matrices.

5

Create concept maps linking eigenvalues, eigenvectors, and diagonalization.

6

Practice past exam papers to familiarize yourself with question formats.

7

Allocate sufficient time for reviewing key theorems and proofs.

8

Form study groups to discuss challenging concepts and problem-solving strategies.

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