Linear Algebra

Linear algebra deals with the study of vectors, matrices, linear functions, and linear equations. Linear algebra is used to solve systems of linear equations and is applied in various fields, including geometry, engineering, and functional analysis. The general form of linear equations is: \(a_1x_1 + a_2x_2 + …. + a_nx_n = b\)
Where a is the coefficient
x is the variable  
b is the constant
n is the number of terms 

In mathematics, linear algebra helps us understand patterns, relationships, and structures, and is applied in fields such as science, engineering, computer science, and data analysis. Some important topics in linear algebra are: 
 

• Euclidean vector spaces: A space consisting of vectors that follow the rules of Euclidean geometry. 
   
• Eigenvalues and eigenvectors: An eigenvalue and its corresponding eigenvector are used to describe how a linear transformation of vectors works. 
   
• Orthogonal matrices: An orthogonal matrix is a square matrix where rows and columns are mutually perpendicular to each other and each has a length of one.  
   
• Linear transformations: A function between vector spaces that preserves vector addition and scalar multiplication. 
   
• Projections: In linear algebra, a projection is the operation of mapping a vector onto another vector. 
   
• Solving systems of equations with matrices: A system of linear equations involves a set of two or more linear equations. To solve them effectively, we use matrix operations such as matrix inversion or the use of the inverse of the coefficient matrix.  
   
• Matrix operations: Using matrices to perform basic operations like addition, multiplication, and scalar multiplication. 
   
• Positive-definite matrices: A positive-definite matrix is a symmetric matrix that, when used in quadratic form (xT Ax), always results in a positive value for any non-zero vector x.  
   
• Singular value decomposition: Is the process that expresses a matrix as the product of three matrices, which are an orthogonal matrix, a diagonal matrix, and the transpose of another orthogonal matrix. 
   
• Linear dependence and independence: Linear dependence and independence are a set of vectors that can be expressed in terms of each other. It is linearly dependent if at least one vector can be written as a combination of the others. If no such combination exists, the vectors are considered linearly independent.  

Linear algebra is often divided into three main branches based on the complexity of topics: elementary, advanced, and applied linear algebra. Here, we will discuss each branch in detail. 
 

• Elementary linear algebra: Elementary linear algebra is introduced to students at a basic level. It includes simple matrix operations, solving systems of linear equations, and understanding the concept of vectors. Some key terms related to linear algebra are: 
   
• Scalars: Quantities that have only magnitude and not direction, and are mostly represented using real numbers. 
   
• Vectors: The elements in a vector space that have both magnitude and direction. 
   
• Vector space: The space of vectors that can be added or multiplied by scalars. 
   
• Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. 
   
• Matrix operations: The matrix operations are the fundamental operations using matrices, including addition, subtraction, and scalar and matrix multiplication.  
   
• Advanced linear algebra: Advanced linear algebra is a more complex concept as compared to elementary linear algebra. It includes topics such as matrices, linear transformations, and abstract vector spaces. Key concepts related to advanced linear algebra are: 
   
• Linear transformation: A linear transformation is a function that maps one vector space to another by preserving vector addition and scalar multiplication. 
   
• Inverse of a matrix: The product of a matrix with its inverse is the identity matrix.  
   
• Eigenvector: A non-zero vector that, when a linear transformation is applied to it, changes only in magnitude and is scaled by a constant called the eigenvalue. 
   
• Linear map: A linear map is a function between two vector spaces that preserves vector addition and scalar multiplication. 
   
• Applied linear algebra: Applied linear algebra uses concepts from both elementary and advanced linear algebra. It involves topics such as QR factorization, Schur’s complement of a matrix, and vector norms.   
   
• Linear algebra formulas
  In linear algebra, formulas are used to simplify expressions and matrices. The formulas can be classified into three categories like linear equations, vectors, and matrices. 
   
• Linear equations: The equations with the highest degree of one are the linear equations, and the common forms and basic properties are: 
  
  \(ax + by = c\\ y = mx + b\\ a + b = b + a\\ a + 0 = 0 + a = a \)
   
• Vectors: For vectors \(\mathbf{u} = (u_1, u_2, u_3) \text{ and } \mathbf{v} = (v_1, v_2, v_3)\) the important formulas are:
  
  \(\mathbf{u} + \mathbf{v} = (u_1 + v_1, \; u_2 + v_2, \; u_3 + v_3) \ \ \ \text{(Vector addition)}\)
  
  \(\mathbf{u} - \mathbf{v} = (u_1 - v_1, \; u_2 - v_2, \; u_3 - v_3) \ \ \text{(Vector subtraction)}\)
  
  \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k} \ \ \text{(Vector in terms of unit vectors)}\)
  
  \(\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \ \ \text{(Dot product)}\)
  
  \(\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix} \ \ \text{(Cross product)}\)
  
  \(\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2} \ \ \text{(Magnitude of } \mathbf{u}) \)
  
  \(\text{Angle between } \mathbf{u} \text{ and } \mathbf{v}: \quad \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}\)
   
• Matrix: The important formulas used in linear algebra related to matrices are: 
  For any matrices A and B with elements \(a_{ij}\) and \(b_{ij}\).
  
  \(A^{-1}A = 1\)
  \(C = A + B\)
  \(C = A - B\)
  \(KA = ka_{ij}\)

Linear Algebra is one of the most powerful and widely used branches of mathematics — essential in engineering, data science, AI, computer graphics, physics, and more. Here are some of the tips and tricks to master linear algebra.
 

1. Try to visualize everything. Linear Algebra is geometric at heart. Try to see what’s happening rather than just calculate.
   
   Vector addition → moving arrows head-to-tail.

   Matrix transformations → stretching, rotating, or flipping shapes.

   Eigenvectors → directions that don’t rotate when transformed.
   
   Use graph paper or online tools (like GeoGebra or Desmos) to visualize matrix transformations.
    

2. Understand inverse matrices conceptually. The inverse of a matrix “undoes” what the matrix does. \(A^{-1}A = I\), where I is the identity matrix.Multiply by a number’s reciprocal. It is as the same idea for matrices.
    

3. Practice row operations and Gaussian elimination. Row operations are the key to solving systems and finding inverses. Three valid operations are as, swap two rows, multiply a row by a nonzero constant, add/subtract a multiple of one row from another
    

4. Learn the language of vector spaces. Understand the key terms.
    

   Span → all combinations of given vectors

   Basis → smallest set that can “build” the space

   Dimension → number of vectors in a basis
    

5. Practice more with some of the real-life applications. Linear Algebra shows up everywhere. Some examples are, computer graphics (rotations and scaling), data science (PCA uses eigenvectors), networks and circuits (solving systems).

Linear algebra is applied in various fields like engineering, data science, computer science, physics, and biology. Below are some real-life applications. 
 

• In signal processing: Linear algebra is used to represent and manipulate audio, image, and video signals using vectors and matrices. For example, SVD is used in image compression to reduce file size while preserving important features and minimizing quality loss. 
   
• In linear programming: Linear algebra is used to optimize a linear objective function subject to linear constraints. This involves solving systems of linear equations and inequalities. 
   
• In computer science and machine learning: Linear algebra is used to build models and perform data analysis. 
   
• Robotics: Robots rely on coordinate transformations to move in space. Matrices represent position, orientation, and motion.
   
• Computer vision and AI: Linear Algebra underlies face recognition, object detection, and image classification. Eigenvectors and singular value decomposition (SVD) are widely used for data compression and feature extraction.