Linear Algebra Notes for GATE 2026 (Handwritten)

Linear Algebra is one of the most important topics in GATE Mathematics, forming the foundation for engineering concepts such as control systems, signal processing, machine learning, and network analysis. A strong command of Linear Algebra helps solve a wide range of numerical and conceptual questions efficiently.

To simplify your preparation, here are the Linear Algebra Notes for GATE 2026 (Handwritten) — structured topic-wise for quick revision, conceptual clarity, and easy memorization.

🧩 Topics Covered in Linear Algebra Notes

1. Vector Space

A vector space is a collection of vectors that can be added together and multiplied by scalars, following specific algebraic rules.

Key points include:

Concept of vectors, addition, and scalar multiplication
Subspaces and their properties
Examples of vector spaces in ℝ², ℝ³, and polynomial functions

Understanding vector spaces builds the foundation for most Linear Algebra problems in GATE.

2. Basis & Dimension

The basis of a vector space is a set of linearly independent vectors that can generate all other vectors in that space through linear combinations.

The dimension of a vector space is the number of vectors in its basis.
Change of basis helps represent vectors in different coordinate systems.

These concepts frequently appear in GATE questions related to coordinate transformations and representation.

3. Linear Dependence & Independence

This topic focuses on determining whether a set of vectors is linearly dependent (one can be expressed as a combination of others) or independent.

Tests for dependence and independence
Applications in solving vector-related problems
Relation with matrix rank and determinant

Understanding this is crucial for simplifying systems of equations and finding matrix solutions.

4. Matrix Algebra

Matrices form the core of Linear Algebra and are widely used in engineering applications.

Topics include:

Matrix operations: addition, subtraction, multiplication, transpose, and inverse
Special matrices: identity, diagonal, symmetric, and orthogonal matrices
Properties and shortcuts for determinants and matrix multiplication.

Mastering these operations helps in solving large-scale linear systems efficiently.

5. Eigenvalues & Eigenvectors

Eigenvalues and eigenvectors represent how a matrix scales or transforms a vector.

Derivation using the characteristic equation ∣A−λI∣=0
Diagonalization of matrices using eigen decomposition
Applications in dynamic systems, control theory, and stability analysis.

This topic often appears in GATE numerical questions involving stability or transformations.

6. Rank of a Matrix

The rank of a matrix represents the number of linearly independent rows or columns.

Row rank and column rank are always equal.
The Rank–Nullity Theorem:
Rank(A)+Nullity(A)=Number of columns of A

This concept helps determine the existence and number of solutions to linear systems — a frequent GATE question type.

7. System of Linear Equations

This section focuses on solving equations of the form AX=B.

Existence and uniqueness of solutions based on matrix rank
Consistent and inconsistent systems
Applications using Gauss elimination and Cramer’s rule

Understanding this helps in determining when a system has a unique solution, infinite solutions, or no solution, which is a classic GATE pattern.

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🧾 Conclusion

Linear Algebra forms the backbone of mathematical analysis and problem-solving in the GATE exam. These handwritten Linear Algebra Notes for GATE 2026 summarize all crucial concepts — from vector spaces to eigenvalues — in a concise, structured manner to help you master every topic step-by-step.

Use these notes for quick revision, last-minute preparation, and conceptual strengthening to secure a high score in GATE 2026.

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