Linear algebra is the study of linear systems of equations and the properties of vectors and matrices. It involves the manipulation of vectors and matrices using operations such as addition, subtraction, scalar multiplication, dot product, cross product, and matrix multiplication. Linear algebra is used extensively in fields such as computer graphics, data science, optimization, and signal processing.
Differential calculus, on the other hand, is concerned with the study of rates of change and slopes of curves. It involves the computation of derivatives, which represent the instantaneous rate of change of a function at a given point. Differential calculus is used in various fields such as physics, engineering, economics, and finance to model real-world phenomena and optimize systems.
Both linear algebra and differential calculus are essential tools for many areas of science and engineering. They are also interconnected, as linear algebra concepts are used in the study of differential equations, which are equations that involve derivatives.
LINEAR ALGEBRA AND DIFFERENTIAL CALCULUS
Syllabus of LADC
UNIT I
VECTOR AND MATRIX ALGEBRA
- Vector space (definition and examples)
- linear independence of vectors
- orthogonality of vectors
- projection of vectors
- Symmetric, Hermitian, skew-symmetric, skew-Hermitian, orthogonal and unitary matrices
- Rank of a matrix by echelon reduction
- Solution of a linear algebraic system of equations (homogeneous and non-homogeneous)
UNIT II
MATRIX EIGENVALUE PROBLEM AND QUADRATIC FORMS
- Determination of eigenvalues and eigenvectors of a matrix
- Properties of eigenvalues and eigenvectors (without proof)
- diagonalization of a matrix
- orthogonal diagonalization of symmetric matrices
- Similarity of matrices
Quadratic Forms:
- Definiteness and nature of a quadratic form
- reduction of quadratic form to canonical form by orthogonal transformation
UNIT III
MATRIX DECOMPOSITION AND PSEUDO INVERSE OF A MATRIX
- Spectral decomposition of a symmetric matrix
- L-U decomposition
- Gram-Schmidt
- orthonormalization of vectors
- Q-R factorization
- Singular value decomposition
- Moore-Penrose pseudo inverse of a matrix
- least squares solution of an over determined system of equations using pseudo inverse
UNIT IV
MULTIVARIABLE DIFFERENTIAL CALCULUS AND FUNCTION OPTIMIZATION
Partial Differentiation:
- Total derivative
- Jacobian
- Functional dependence
- Unconstrained optimization of functions using the Hessian matrix
- constrained optimization using Lagrange multiplier method
UNIT V
SINGLE VARIABLE CALCULUS
Mean value theorems:
- Rolle’s Theorem
- Lagrange’s Mean value theorem and Taylor’s theorem (without proof)
- their geometrical interpretation
- approximation of a function by Taylor’s series
- Applications of definite integrals to evaluate surface areas and volumes of revolutions of curves (for Cartesian coordinates)
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