MTH1004 Linear Algebra

Module Co-ordinator: Dr Helen Christodoulidi ([email protected])

MTH1004 Linear Algebra Question-Based Revision

Course Components

• Portfolio (40%)
  • Web Assignments (15%)
  • In-class Test (25%)
• Final Exams (60%)

Current percentage (rounded up): 15%

Learning Outcomes

• LO1 Formulate the connection between linear transformations and their matrices in different bases.
• LO2 Find orthogonal bases and complements; find inverses of orthogonal matrices.
• LO3 Find kernel, range, rank and nullity of a linear transformation.
• LO4 Find eigenvalues and eigenvectors; apply them to diagonalization of matrices and finding functions of linear mappings and matrices.
• LO5 Diagonalize quadratic forms by using orthogonal diagonalization of symmetric matrices

Summary Content

MTH1004 Linear Algebra Summary Notes - a comprehensive guide to solve any previously seen questions. MTH1004 Linear Algebra Summary Questions - a compilation of all questions from lectures, practicals, and assignments. MTH1004 Linear Algebra Practice Tests - a list of past, practice, and predicted papers; the latter generated by me.

Flashcards

Optionally, you can take the most important flashcards from here and make them physical for more tactile revision.

MTH1004 Linear Algebra Flashcards - a document full of flashcards to practice key definitions, theorems, and methods; includes an Anki deck of the same flashcards in .apkg format.

Notes

1. Vectors

1.1 Vectors in the Plane

• Vector Representation: Representing vectors in 2D space using components.
• Vector Operations: Addition, subtraction, and scalar multiplication of vectors.
• Magnitude and Direction: Understanding the length and direction of vectors.

1.2 Vectors in Higher Dimensions

• Dot Product: Definition and properties of the dot product in various dimensions.
• Angle between Vectors: Calculating the angle between vectors in space.
• Vector Length: Determining the length of vectors in higher dimensions.

2. Linear Systems

2.1 Solving Linear Systems

• Systems of Equations: Representing linear systems of equations.
• Gaussian Elimination: Method for solving systems of linear equations.
• Matrix Form: Expressing linear systems in matrix form for solving.

3. Matrices

3.1 Matrix Operations

• Matrix Addition and Subtraction: Performing addition and subtraction operations on matrices.
• Matrix Multiplication: Understanding the rules and properties of matrix multiplication.
• Inverse Matrix: Finding the inverse of a matrix.

3.2 Matrix Rank

• Rank of a Matrix: Defining the rank of a matrix and its significance in linear algebra.

4. Vector Spaces

4.1 Vector Spaces and Bases

• Vector Space Properties: Exploring the properties of vector spaces.
• Basis Vectors: Understanding basis vectors and their role in vector space.

4.2 Dimension of Vector Spaces

• Space Dimension: Defining the dimension of a vector space and its implications.

5. Eigenvalues and Eigenvectors

5.1 Eigenvalues and Eigenvectors

• Eigenvalue Equation: Solving for eigenvalues and eigenvectors of a matrix.
• Eigenspaces: Understanding eigenspaces associated with eigenvalues.

6. Linear Transformations

6.1 Transformations in Vector Spaces

• Linear Maps: Defining linear transformations between vector spaces.

7. Change of Basis

7.1 Basis Transformation

• Change of Basis Matrix: Finding the matrix to transition between different bases.
• Orthogonal Bases: Exploring orthogonal and orthonormal bases.

8. Orthogonal Matrices and Quadratic Forms

8.1 Orthogonal Matrices

• Properties of Orthogonal Matrices: Characteristics and properties of orthogonal matrices.
• Quadratic Forms: Understanding quadratic forms and their representation.

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Linear Algebra Mid-Term Notes

Linear Algebra Mid-Term Prediction | Linear Algebra Mid-Term Prediction (Solutions) | Linear Algebra Mid-Term Prediction (Cheat Sheet)

Note: Can bring a basic calculator for arithmetic, 100 marks total over one hour.

• Eigenvalues and eigenvectors of a matrix - conditions for diagonalisation.
• Solving linear (homogenous) systems - $\mathbf{A}x=0⟹\text{Null}(A)=\text{Vector Subspace}=\text{Span}(\text{basis})$, using Gauss-Jordan elimination.
• Inverse of a matrix, using Gauss-Jordan elimination.
• Find the basis of a vector space and prove that some vectors form a basis, e.g. $3$ given vectors in ${\mathbb{R}}^3$ form a basis.