## PHY 222: Linear Algebra

3 credits | Prerequisites: MAT 104

### Course rationale

Linear algebra is an important branch of mathematics that finds many applications in physics. Starting with the system of linear equations this course develops the theory of matrices. This further goes to the theory of vector space and operators. Giving more structures to vector space this course introduces inner product space and the theory of linear transformations. The contents of this course find many applications in physics including Quantum Mechanics, Relativity, and others.

### Course content

Row Operations: a system of linear equations, row reduction, Gaussian elimination; Matrix: definition, matrix algebra, inverse matrix; Determinant: definition, cofactor expansion, Cramer’s rule; System of Linear Equations: matrix and determinant methods for finding the solution, overdetermined and underdetermined solutions; Vectors in Rn and Cn: a review of geometric vectors in R2 and R3 space, vectors in Rn and Cn, inner product, norm and distance in Rn and Cn; Vector Space: linear vector space, subspace; Linear Dependence and Independence: basis and dimension of vector spaces, row and column space of a matrix, the rank of matrices, solution spaces of systems of linear equations; Eigenvalues and Eigenvectors: characteristic equation, diagonalization, Cayley-Hamilton theorem, applications; General Linear Transformation: kernel and image of a linear transformation and their properties, matrix representation of linear transformations, change of bases, isomorphism, similarity; Inner Product Space: inner product, angle, and orthogonality, Gram-Schmidt procedure

### Course objectives

- Identify and solve systems of linear equations.
- Use matrix and determinant to solve the system of linear equations.
- Represent linear transformations by appropriate matrices.
- Extend the usual idea of vector space into abstract vector space.
- Calculate eigenvalues and eigenvectors of matrices.
- Understand the general linear transformation and represent them by an appropriate matrix
- Extend the idea of the dot product of vector analysis to a more general notion of inner product, and inner product space.

### References

- Elementary Linear Algebra (12th edition) by Howard Anton, and Anton Kaul
- Linear Algebra with Applications (8th edition) by Steve Leon
- Introduction to Linear Algebra (5th edition) by Gilbert Strang
- Linear Algebra Done Right (3rd edition) by Sheldon Axler
- Linear Algebra (5th edition) by Stephen Friedberg, Arnold Insel, and Lawrence Spence