The LAB - Overview of Linear Algebra Concepts with Python Labs This comprehensive lab guide explores foundational and advanced linear algebra concepts using Python and libraries such as NumPy, SciPy, SymPy, Matplotlib, and NetworkX. Emphasis is placed on both theoretical explanations and practical coding exercises to build intuition and skills. --- Chapter 1. Vectors, Scalars, and Geometry Scalars vs Vectors: Scalars are single numbers; vectors are lists of numbers representing points or directions in space. Vector notation: Vectors have symbols, components, and can be visualized as arrows. Vector operations: Addition and scalar multiplication are component-wise. Linear combinations build new vectors from scalings and sums. Span: The set of all linear combinations of given vectors. Span can be a line or a plane depending on independence. Length (Norm) and Distance: Measured using Pythagorean theorem; distance between vectors is norm of their difference. Dot Product: Combines magnitudes and angles; computed algebraically and geometrically relates to cosine of angle. Angles between Vectors: Calculated with dot product; cosine similarity is widely used in machine learning. Projections and Decompositions: Split a vector into components parallel and perpendicular to another vector. Extends to higher dimensions. Inequalities: Cauchy–Schwarz bounds dot product; triangle inequality relates vector norms. Orthonormal Sets: Collections of mutually perpendicular unit vectors, forming "nice" bases. --- Chapter 2. Matrices and Basic Operations Matrices as tables and machines: Matrices store numbers and also act as linear transformations. Matrix shapes, indexing, and blocks: Understanding dimensions, slicing rows/columns, and submatrices. Matrix addition and scalar multiplication: Performed element-wise; shapes must match. Matrix–Vector product: Result is a new vector formed by linear combinations of matrix columns. Matrix–Matrix product: Composition of linear transformations; associative but not commutative. Identity, Inverse, Transpose: Identity matrix acts like 1 in multiplication. Inverse reverses matrix effect if it exists. Transpose swaps rows and columns. Special matrices: Symmetric (equal to transpose). Diagonal (nonzero entries only on diagonal). Triangular (upper or lower zeros). Permutation (rearranges vector components). Trace: Sum of diagonal entries; linear and invariant under similarity transforms. Affine transforms and homogeneous coordinates: Allow translations via augmented dimension; unify all transforms in one matrix. Computing with matrices: Includes cost analysis and simple performance tips using vectorization and data types. --- Chapter 3. Linear Systems and Elimination From equations to matrices: Linear systems can be expressed as \(A x = b\). Row operations: Swap rows, scale rows, replace rows; preserve solution set and facilitate elimination. Row-echelon and reduced row-echelon forms: Standard matrix forms simplifying solution extraction. Pivots, free variables: Identify basic and free variables from matrix form; key to solutions. Solving systems: Unique solution: pivots in all variable columns. Infinite solutions: free variables remain. No solutions: inconsistency detected (e.g., \(0 = c\) with \(c \neq 0\)). Gaussian elimination: Stepwise elimination leading to back substitution. Back substitution: Solve upper-triangular system for unknowns. Rank: Number of pivots; relates to solution properties. LU factorization: Decompose \(A = L U\) matrices from elimination; efficient solving repeated linear systems. --- Chapter 4. Vector Spaces and Subspaces Vector spaces: Sets with addition and scalar multiplication satisfying specific axioms (closure, associativity,