Linear Algebra
MA25C02 2nd Semester | 2025 Regulation
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Subject: Linear Algebra
2025 regulation - 2nd semester for All Department including EEE, ECE, MECH, Civil, CSE, IT, AID Department, etc. Subject Code: MA25C02 , Subject Name: Linear Algebra, Batch: 2025, 2026, 2027, 2028. Institute: Anna University Affiliated Engineering College, TamilNadu. This page has Linear Algebra or M2 study material, notes, semester question paper pdf download, important questions, lecture notes.
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Linear Algebra
- Introduction of Vector Spaces
- Definition of Vector Spaces
- Vector Spaces: Basic Example Problems
- Vector Spaces: Theorems Part 1
- Vector Spaces: Theorems Part 1: Example Solved Problems
- Definition and Theorem of Subspaces
- Subspaces (Vector Spaces): Example Solved Problems
- Vector Subspaces: Theorems Part 2
- Vector SubSpaces: Theorems Part 2 - Example Solved Problems
- Linear Combination
- Linear Combination: Example Solved Problems
- Linear span
- Linear span: Example Solved Problems
- Vector Spaces: Theorems Part 3
- Vector Spaces: Theorems Part 3 - Example Solved Problems
- Vector Spaces: Theorems Part 4
- Vector Spaces: Theorems Part 4 - Example Solved Problems
- Vector Spaces: Theorems Part 5
- Definition of Linear Dependence and Linear Independence
- Linear Dependence and Linear Independence: Example Solved Problems
- Vector Spaces: Theorems Part 6
- Vector Spaces: Theorems Part 6 - Example Solved Problems
- Bases and Dimensions
- Bases and Dimensions: Example Solved Problems
- Vector Spaces: Theorems Part 7
- Vector Spaces: Theorems Part 7 - Example Solved Problems
- Vector Spaces - Theorem Part 8
- Vector Spaces: Exercise Problems
- Vector Spaces: 2 Marks Important Questions with Answer
- Vector Spaces: Multiple Choice Questions and Answers
- Linear Transformation and Diagonalization
- Definition and Properties of Linear Transformation
- Linear Transformation: Example Solved Problems
- Null space (or) Kernel N(T) and Range (or) Image R(T)
- Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems
- Null space (or) Kernel N(T) and Range (or) Image R(T): Theorem Part 2
- Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems - Part 2
- Linear Transformations: Projection
- Linear Transformation: Example Solved Problems
- Matrix Representation of a Linear Transformation
- Matrix Representation of a Linear Transformation: Example Solved Problems
- Matrix Representation of a Linear Transformation: Theorems
- Matrix Representation of a Linear Transformation: Example Solved Problems Using Theorems
- Definition of Diagonalizable, Eigenvalues and Eigenvector
- Eigenvalues and Eigenvectors: Theorem 1
- Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems
- Eigenvalues and Eigenvectors: Theorem 2
- Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems
- Linear Transformations: Characteristic polynomial
- Characteristic Polynomial: Example Solved Problems
- Eigenvalues and Eigenvectors: Theorem 3
- Eigenvalues and Eigenvectors: Theorem 3 - Example Solved Problems
- Diagonalizability: Theorems Part 1
- Diagonalizability: Theorems Part 1 - Example Solved Problems
- Diagonalizability: Theorems Part 2
- Diagonalizability: Theorems Part 2 - Example Solved Problems
- System of Differential Equations
- Linear Transformation and Diagonalization: Exercise Questions
- Linear Transformation and Diagonalization: 2 Marks Important Questions with Answer
- Linear Transformation and Diagonalization: Multiple Choice Questions and Answers
- Introduction of Inner Product Spaces
- Inner Products and Norms: Definition, Theorem
- Inner Products and Norms: Example Solved Problems - Part 1
- Definitions of Unit vector, Orthogonal, Orthonormal
- Inner Products and Norms: Example Solved Problems - Part 2
- The Gram Schmidt Orthogonalization: Theorems Part 1
- The Gram Schmidt Orthogonalization: Theorems Part 1 - Example Solved Problems
- The Gram Schmidt Orthogonalization: Theorem Part 2
- The Gram Schmidt Orthogonalization: Theorems Part 2 - Example Solved Problems
- The Gram Schmidt Orthogonalization: Theorem Part 3
- The Gram Schmidt Orthogonalization: Theorem Part 3 - Example Solved Problems
- The Gram Schmidt Orthogonalization: Theorems Part 4
- The Gram Schmidt Orthogonalization: Theorems Part 4 - Example Solved Problems
- Adjoint of Linear Operator
- Adjoint of Linear Operator: Example Solved Problems
- Least Squares Approximation
- Least Squares Approximation: Example Solved Problems
- Inner Product Spaces: Exercise
- Inner Product Spaces : 2 Marks Important Questions With Answer
- Inner Product Spaces : Multiple Choice Questions and Answers
- Matrix Decomposition: Inner Products and Norms
- Matrix Decomposition: Example Solved Problems
- Orthogonal Transformation of a Symmetric Matrix to Diagonal Form
- Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Example Solved Problems
- Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Exercise Problem Questions
- Positive Definite Matrices
- Positive Definite Matrices: Example Solved Problems
- Matrix Decomposition: Positive Definite Matrices: EXERCISE Questions
- QR Decomposition
- QR Decomposition: Example Solved Problems
- Singular Value Decomposition
- Singular Value Decomposition: Example Solved Problems
- Least Square Solutions
- Least Square Solutions: Example Solved Problems
- QR Decomposition, Singular Value Decomposition, Least Square Solutions: Exercise Problems
- Matrix Decomposition: 2 Marks Important Questions with Answer
- Matrix Decomposition: Multiple Choice Questions and Answers
- Computation of Linear Dependence and Independence by Using Python (Linear Algebra: Vector Spaces)
- Computation of Linear Dependent and Independence By Using C (Linear Algebra: Vector Spaces)
- Computation of Span and Basis of Vectors by Using Python (Linear Algebra: Vector Spaces)
- Computation of Span and Basis of Vectors by Using C (Linear Algebra: Vector Spaces)
- Computation of Dimensions of Subspaces in Vectors by Using Python (Linear Algebra: Vector Spaces)
- Computation of Dimension of Subspaces by Using C (Linear Algebra: Vector Spaces)
- Computation of Null Space and Range of a Matrix by Using Python (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Null Space and Range of Matrix by Using C (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Eigen Values and Eigen Vectors Using Python (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Eigenvalues and Eigenvectors Using C (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Inner Products and Vector Norms by Using Python (Linear Algebra: Inner Product Spaces)
- Computation of Inner Products and Vector Norms by Using C (Linear Algebra: Inner Product Spaces)
- Computation of QR Decomposition by Using Python (Linear Algebra: Matrix Decomposition)
- Computation of QR Decomposition by Using C (Linear Algebra: Matrix Decomposition)
- Computation of SVD by Using Python (Linear Algebra: Matrix Decomposition)
- Computation of SVD by Using C (Linear Algebra: Matrix Decomposition)
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Linear Algebra
UNIT I: Vector Spaces
- Introduction of Vector Spaces
- Definition of Vector Spaces
- Vector Spaces: Basic Example Problems
- Vector Spaces: Theorems Part 1
- Vector Spaces: Theorems Part 1: Example Solved Problems
- Definition and Theorem of Subspaces
- Subspaces (Vector Spaces): Example Solved Problems
- Vector Subspaces: Theorems Part 2
- Vector SubSpaces: Theorems Part 2 - Example Solved Problems
- Linear Combination
- Linear Combination: Example Solved Problems
- Linear span
- Linear span: Example Solved Problems
- Vector Spaces: Theorems Part 3
- Vector Spaces: Theorems Part 3 - Example Solved Problems
- Vector Spaces: Theorems Part 4
- Vector Spaces: Theorems Part 4 - Example Solved Problems
- Vector Spaces: Theorems Part 5
- Definition of Linear Dependence and Linear Independence
- Linear Dependence and Linear Independence: Example Solved Problems
- Vector Spaces: Theorems Part 6
- Vector Spaces: Theorems Part 6 - Example Solved Problems
- Bases and Dimensions
- Bases and Dimensions: Example Solved Problems
- Vector Spaces: Theorems Part 7
- Vector Spaces: Theorems Part 7 - Example Solved Problems
- Vector Spaces - Theorem Part 8
- Vector Spaces: Exercise Problems
- Vector Spaces: 2 Marks Important Questions with Answer
- Vector Spaces: Multiple Choice Questions and Answers
UNIT II: Linear Transformations and Diagonalization
- Linear Transformation and Diagonalization
- Definition and Properties of Linear Transformation
- Linear Transformation: Example Solved Problems
- Null space (or) Kernel N(T) and Range (or) Image R(T)
- Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems
- Null space (or) Kernel N(T) and Range (or) Image R(T): Theorem Part 2
- Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems - Part 2
- Linear Transformations: Projection
- Linear Transformation: Example Solved Problems
- Matrix Representation of a Linear Transformation
- Matrix Representation of a Linear Transformation: Example Solved Problems
- Matrix Representation of a Linear Transformation: Theorems
- Matrix Representation of a Linear Transformation: Example Solved Problems Using Theorems
- Definition of Diagonalizable, Eigenvalues and Eigenvector
- Eigenvalues and Eigenvectors: Theorem 1
- Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems
- Eigenvalues and Eigenvectors: Theorem 2
- Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems
- Linear Transformations: Characteristic polynomial
- Characteristic Polynomial: Example Solved Problems
- Eigenvalues and Eigenvectors: Theorem 3
- Eigenvalues and Eigenvectors: Theorem 3 - Example Solved Problems
- Diagonalizability: Theorems Part 1
- Diagonalizability: Theorems Part 1 - Example Solved Problems
- Diagonalizability: Theorems Part 2
- Diagonalizability: Theorems Part 2 - Example Solved Problems
- System of Differential Equations
- Linear Transformation and Diagonalization: Exercise Questions
- Linear Transformation and Diagonalization: 2 Marks Important Questions with Answer
- Linear Transformation and Diagonalization: Multiple Choice Questions and Answers
UNIT III: Inner Product Spaces
- Introduction of Inner Product Spaces
- Inner Products and Norms: Definition, Theorem
- Inner Products and Norms: Example Solved Problems - Part 1
- Definitions of Unit vector, Orthogonal, Orthonormal
- Inner Products and Norms: Example Solved Problems - Part 2
- The Gram Schmidt Orthogonalization: Theorems Part 1
- The Gram Schmidt Orthogonalization: Theorems Part 1 - Example Solved Problems
- The Gram Schmidt Orthogonalization: Theorem Part 2
- The Gram Schmidt Orthogonalization: Theorems Part 2 - Example Solved Problems
- The Gram Schmidt Orthogonalization: Theorem Part 3
- The Gram Schmidt Orthogonalization: Theorem Part 3 - Example Solved Problems
- The Gram Schmidt Orthogonalization: Theorems Part 4
- The Gram Schmidt Orthogonalization: Theorems Part 4 - Example Solved Problems
- Adjoint of Linear Operator
- Adjoint of Linear Operator: Example Solved Problems
- Least Squares Approximation
- Least Squares Approximation: Example Solved Problems
- Inner Product Spaces: Exercise
- Inner Product Spaces : 2 Marks Important Questions With Answer
- Inner Product Spaces : Multiple Choice Questions and Answers
UNIT IV: Matrix Decomposition
- Matrix Decomposition: Inner Products and Norms
- Matrix Decomposition: Example Solved Problems
- Orthogonal Transformation of a Symmetric Matrix to Diagonal Form
- Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Example Solved Problems
- Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Exercise Problem Questions
- Positive Definite Matrices
- Positive Definite Matrices: Example Solved Problems
- Matrix Decomposition: Positive Definite Matrices: EXERCISE Questions
- QR Decomposition
- QR Decomposition: Example Solved Problems
- Singular Value Decomposition
- Singular Value Decomposition: Example Solved Problems
- Least Square Solutions
- Least Square Solutions: Example Solved Problems
- QR Decomposition, Singular Value Decomposition, Least Square Solutions: Exercise Problems
- Matrix Decomposition: 2 Marks Important Questions with Answer
- Matrix Decomposition: Multiple Choice Questions and Answers
Practice Programs in Python and C Languages
- Computation of Linear Dependence and Independence by Using Python (Linear Algebra: Vector Spaces)
- Computation of Linear Dependent and Independence By Using C (Linear Algebra: Vector Spaces)
- Computation of Span and Basis of Vectors by Using Python (Linear Algebra: Vector Spaces)
- Computation of Span and Basis of Vectors by Using C (Linear Algebra: Vector Spaces)
- Computation of Dimensions of Subspaces in Vectors by Using Python (Linear Algebra: Vector Spaces)
- Computation of Dimension of Subspaces by Using C (Linear Algebra: Vector Spaces)
- Computation of Null Space and Range of a Matrix by Using Python (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Null Space and Range of Matrix by Using C (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Eigen Values and Eigen Vectors Using Python (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Eigenvalues and Eigenvectors Using C (Linear Algebra: Linear Transformation and Diagonalization)
- Computation of Inner Products and Vector Norms by Using Python (Linear Algebra: Inner Product Spaces)
- Computation of Inner Products and Vector Norms by Using C (Linear Algebra: Inner Product Spaces)
- Computation of QR Decomposition by Using Python (Linear Algebra: Matrix Decomposition)
- Computation of QR Decomposition by Using C (Linear Algebra: Matrix Decomposition)
- Computation of SVD by Using Python (Linear Algebra: Matrix Decomposition)
- Computation of SVD by Using C (Linear Algebra: Matrix Decomposition)
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MA25C02:
Linear Algebra
Course Objectives:
•
To impart foundational knowledge in linear algebra essential for analysing and
solving problems in engineering applications.
•
To provide the knowledge on computation using software and interpret key linear
algebra concepts using software.
Vector
Spaces Introduction to Vector Spaces, Examples, Subspaces,
Linear Combinations, Span, Generating Sets, Linear Dependence and Independence,
Basis and Dimension, Dimension of Subspaces.
Activities:
Open-Source software, exercises to test linear dependence and independence
using rank, compute span and basis of a set of vectors, determine the dimension
of subspaces, and illustrate the concept of subspace and basis in 𝑹𝟐/𝑹𝟑
with visualization.
Linear
Transformations and Diagonalization: Null space, Range,
Dimension Theorem (statement only), Matrix representation of a linear
transformation, Eigenvalues & Eigenvectors, Diagonalizability.
Activities:
Open-Source software, exercises to compute the matrix representation of a
linear transformation, find the null space and range of a matrix, and compute
eigenvalues and eigenvectors of a matrix.
Inner
Product Spaces:Inner product, Norms, Cauchy, Schwarz
inequality, Gram, Schmidt orthogonalization, Simple problems (up to 𝑹𝟑).
Activities:
Open-Source software, exercises to compute inner products and vector norms.
Matrix
Decomposition: Orthogonal transformation of a
symmetric matrix to diagonal form
-
Positive definite matrices, QR decomposition, Singular Value Decomposition
(SVD), Least squares solutions- simple problems (up to 3 × 3 matrices )
Activities:
Open-Source software, exercises to check if a matrix is positive definite,
perform QR decomposition and SVD using built-in functions.
Weightage:
Continuous Assessment: 40%, End Semester Examinations: 60%.
Methodology for
Continuous Assessment: Assignment (20%), Software
activity (20%), Quiz (20%), Internal Examinations (40%).
References:
1.
Friedberg, S. H., Insel, A. J., & Spence, L. E. (2022). Linear algebra.
Pearson.
2.
Lay, D. C., Lay, S. R., & McDonald, J. J. (2020). Linear algebra and its
applications with MATLAB. Pearson.
3.
Bronson, R. (2011). Schaum’s outline of matrix operations. McGraw-Hill
Education.
4.
Strang, G., &
Thomson, R. (2005).
Linear algebra and
its applications. Brooks/Cole.
5.
Lipschutz, S., & Lipson, M. (2009). Schaum's outline of linear algebra.
McGraw- Hill.
6.
Kreyszig, E. (2018). Advanced engineering mathematics. Wiley India.
Linear Algebra: UNIT I: Vector Spaces,, Linear Algebra: UNIT II: Linear Transformations and Diagonalization,, Linear Algebra: UNIT III: Inner Product Spaces,, Linear Algebra: UNIT IV: Matrix Decomposition,, Linear Algebra: Practice Programs in Python and C Languages : MA25C02 2nd Semester | 2025 Regulation Linear Algebra
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Subject: Linear Algebra
