MATHS-2: Mathematics for Data Science 2

Topics covered

• Vectors

  • Definition of vectors
  • Vectors in 2D and 3D
  • Vector addition and scalar multiplication
  • Zero vector and negative vectors

• Matrices

  • Definition and types of matrices
  • Matrix addition and scalar multiplication
  • Matrix multiplication
  • Transpose of a matrix
  • Matrix properties

• Systems of Linear Equations

  • Linear equations in two or more variables
  • Matrix representation of a system
  • Coefficient matrix and augmented matrix
  • Geometric interpretation of solutions

• Determinants — Part 1

  • Determinants of (2×2) matrices
  • Determinants of (3×3) matrices
  • Properties of determinants

• Determinants — Part 2

  • Cofactor expansion
  • Minors and cofactors
  • Determinant calculations

Expected skills: Matrix operations, calculating determinants, representing systems of equations using matrices.

Topics covered

• Determinants — Part 3

  • Further determinant properties
  • Determinants and invertibility

• Cramer's Rule

  • Solving systems using determinants
  • Conditions for unique solutions

• Solutions to a system with an invertible coefficient matrix

  • Invertible matrices
  • Matrix inverse
  • Unique solution using the inverse

• Echelon Form

  • Row echelon form
  • Pivot positions
  • Leading entries

• Row Reduction

  • Elementary row operations
  • Reduced row echelon form

• Gaussian Elimination

  • Solving systems of linear equations
  • Identifying inconsistent systems
  • Identifying infinitely many solutions

Expected skills: Solve systems efficiently, use Gaussian elimination and determine the number of solutions.

Topics covered

• Introduction to Vector Spaces

  • Definition of vector spaces
  • Examples of vector spaces
  • Vector addition and scalar multiplication
  • Zero vector and additive inverses

• Properties of Vector Spaces

  • Closure properties
  • Vector space axioms
  • Subspaces
  • Testing whether a set is a subspace

• Linear Dependence

  • Linear combinations
  • Linear dependence of vectors
  • Expressing a vector as a combination of others

• Linear Independence — Part 1

  • Definition and basic tests
  • Homogeneous systems

• Linear Independence — Part 2

  • Independence using row reduction
  • Dependencies among vectors
  • Examples and applications

Expected skills: Determine whether a set forms a vector space or subspace, and test linear dependence and independence.

Topics covered

• What Is a Basis for a Vector Space?

  • Spanning sets
  • Linear independence
  • Definition of a basis

• Finding Bases for Vector Spaces

  • Basis of a subspace
  • Extracting independent vectors
  • Finding a basis using row reduction

• Rank and Dimension

  • Dimension of a vector space
  • Rank of a matrix
  • Relationship between basis and dimension

• Rank and Dimension Using Gaussian Elimination

  • Pivot columns
  • Finding matrix rank
  • Identifying independent columns

Expected skills: Find bases, determine dimensions and calculate matrix rank.

Topics covered

• Rank and Nullity of a Matrix

  • Null space
  • Nullity
  • Finding a basis for the null space — Part 1
  • Finding a basis for the null space — Part 2
  • Rank-nullity theorem

• Introduction to Linear Mapping

  • Linear mappings — Part 1
  • Linear mappings — Part 2
  • Linear transformations
  • Linearity conditions
  • Examples and counterexamples

Expected skills: Find null spaces, calculate nullity and understand the relationship between rank and nullity.

Topics covered

• Linear Transformations, Ordered Bases and Matrices

  • Matrix representation of transformations
  • Ordered bases
  • Transformation matrices

• Image and Kernel of Linear Transformations

  • Kernel of a transformation
  • Image or range of a transformation
  • Kernel and image as subspaces

• Finding Bases for Kernel and Image

  • Calculating the kernel
  • Calculating the image
  • Finding bases for both
  • Rank and nullity of transformations

Expected skills: Represent linear transformations using matrices and determine their kernels, images, bases, ranks and nullities.

Topics covered

• Equivalent and Similar Matrices

  • Equivalence of matrices
  • Similarity of matrices
  • Matrix transformations and relationships

• Affine Subspaces and Affine Mappings

  • Affine subspaces
  • Affine mappings
  • Differences between linear and affine mappings

• Lengths and Angles

  • Length of vectors
  • Distance between vectors
  • Angles between vectors

• Inner Products and Norms

  • Inner product definition
  • Dot product
  • Properties of inner products
  • Norms and distances
  • Examples of inner products on vector spaces

Expected skills: Understand matrix equivalence, calculate lengths and angles, and work with inner products and norms.

Topics covered

• Orthogonality and Linear Independence

  • Orthogonal vectors
  • Orthogonal sets
  • Relationship between orthogonality and independence

• Orthonormal Basis

  • Normalization of vectors
  • Orthonormal sets
  • Orthonormal basis

• Projections Using Inner Products

  • Projection of one vector onto another
  • Orthogonal projections
  • Projection calculations

• Gram-Schmidt Process

  • Orthogonalization
  • Step-by-step Gram-Schmidt algorithm
  • Constructing orthonormal bases

• Orthogonal Transformations and Rotations

  • Orthogonal matrices
  • Properties of orthogonal transformations
  • Rotations in 2D and 3D

Expected skills: Apply Gram-Schmidt, calculate projections and construct orthonormal bases.

Topics covered

• Multivariable Functions: Visualization

  • Functions of two and three variables
  • Domain and range
  • Surfaces and contour plots

• Partial Derivatives

  • Partial differentiation with respect to individual variables
  • First-order partial derivatives
  • Geometric interpretation

• Directional Derivatives

  • Directional rates of change
  • Derivatives along specified directions

• Limits for Scalar-Valued Multivariable Functions

  • Limits in two variables
  • Different paths toward a point
  • Existence and non-existence of limits

• Continuity for Multivariable Functions

  • Continuity at a point
  • Continuity in a domain

• Directional Derivatives in Terms of the Gradient

  • Gradient vector
  • Gradient and directional derivatives
  • Dot product representation

Expected skills: Compute partial derivatives and directional derivatives, and test limits and continuity for multivariable functions.

Topics covered

• Direction of Steepest Ascent and Descent

  • Gradient vector
  • Direction of maximum increase
  • Direction of maximum decrease
  • Magnitude of the gradient

• Tangents for Scalar-Valued Multivariable Functions

  • Tangent lines
  • Linear approximation
  • Tangent planes

• Finding Tangent Hyperplanes

  • Tangent plane equation
  • Linearization of multivariable functions

• Critical Points for Multivariable Functions

  • First-order partial derivatives
  • Gradient equal to zero
  • Finding critical points
  • Candidate points for local extrema

Expected skills: Find gradients, determine steepest directions, calculate tangent planes and locate critical points.

Topics covered

• Higher-Order Partial Derivatives and Hessian Matrix

  • Second-order partial derivatives
  • Mixed partial derivatives
  • Hessian matrix

• Hessian Matrix and Local Extrema for f(x,y)

  • Local maxima and minima
  • Second derivative test for two variables
  • Saddle points

• Hessian Matrix and Local Extrema for f(x,y,z)

  • Critical points in three variables
  • Hessian-based analysis

• Differentiability for Multivariable Functions

  • Differentiability
  • Linear approximation
  • Relationship between differentiability and continuity

• Review of Maths-2

  • Linear algebra revision
  • Multivariable calculus revision
  • Practice and consolidation

Expected skills: Use Hessian matrices to classify critical points and understand differentiability in several variables.