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This document contains questions on linear equations, eigen values and eigen vectors, Fourier series, partial differential equations, and more. It is for the Mathematics II course for II-B.Tech. I-Semester Supplementary Examinations.
What's inside
- Linear equations and their solutions
- Eigen values and eigen vectors
- Fourier series and transforms
- Partial differential equations
- Matrix operations and properties
Who it is for: Students taking the Mathematics II course for II-B.Tech. I-Semester Supplementary Examinations. (Undergraduate)
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Summary
Mathematics II covers linear equations, eigenvalues and eigenvectors, matrix properties, and Fourier series. Students get practice with problems on these topics.
Linear Equations
Linear equations are investigated for consistency and unique solutions. Values of variables are found for which systems of linear equations have no solution, a unique solution, or an infinite number of solutions.
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of matrices are determined. Properties of eigenvalues and eigenvectors are proven. Eigenvalues and eigenvectors of matrices formed from other matrices are found.
Matrix Properties
Matrices are shown to be symmetric, orthogonal, or Hermitian. Inverses of matrices are computed. Quadratic forms are identified. Orthogonal transformations are found.
Fourier Series
Functions are expanded as Fourier series. Fourier sine and cosine series are found. Fourier transforms are computed. Fourier integral theorem is stated and proven.
Key terms
- Eigenvalue
- A scalar by which a matrix can be multiplied to produce another matrix with the same eigenvectors.
- Eigenvector
- A vector that changes at most by a scalar factor when that linear transformation is applied to it.
- Fourier series
- A way to represent a periodic function as a (possibly infinite) sum of sine and cosine waves with different frequencies.
- Orthogonal matrix
- A square matrix whose rows and columns are orthonormal vectors.
- Symmetric matrix
- A square matrix that is equal to its transpose.
- Hermitian matrix
- A square matrix that is equal to its conjugate transpose.
Questions this document helps with
- How do you find the values of variables for which a system of linear equations is consistent?
- What are the properties of eigenvalues and eigenvectors?
- How do you expand a function as a Fourier series?
Summary written with AI from the document's text.
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