דר' דימטרי גוטמן - אלגברה לינארית לפיסיקאים 86-153

 

א. מטרות הקורס (מטרות על / מטרות ספציפיות): אלגברה לינארית – קורס בסיסי במתמטיקה המיועד לסטודנטי שנה ראשונה, תואר ראשון. ידע בקורס זה דרוש כדי ללמוד את כל קורס במתמטיקה או בפיסיקה, ובמיוחד את תורת הקוונטים.

 

ב. תוכן הקורס:

        תכנית הוראה מפורטת לכל השיעורים:

 

Vectors in plane and in space. Addition and scalar multiplication. Norm (length) of vectors. Basics of analytic geometry: equations of line and plane, distance from a point to a line or plane.

 

Linear (vector) spaces. Examples of vector spaces. Subspaces. Linear dependence and independence.

 

Systems of linear equations. Equivalence between systems of linear equations and operations with vectors. Matrices. Systems in triangular and echelon form. Homogeneous systems of linear equations. General solution as a sum of a particular solution and the solution of the corresponding homogeneous system. Theorem of existence of non-zero solution for the homogeneous system where m (number of columns) is greater than n (number if lines).

 

Linear dependence of vortices as a problem of existence of non-zero solution for the homogeneous system of equations. Linear independence in 2D and 3D. Vector (cross) product and its representation as a determinant. Linear spans of vectors. Basis. Dimension. Change of basis. Solutions of the homogeneous system of equation as a linear space and its basis. Isomorphism of vector spaces of the same dimensionality.

 

Inner (scalar) product. Orthogonality. Orthogonal and orthonormal bases. Projections. Gram-Schmidt orthogonalization process. Complex inner product spaces.

 

Transpositions and permutations. Determinants. Calculation of determinants. Determinant and volume. Minors. Rank of matrix. Vector product (in 3D) and generalized vector product.

 

Mappings (functions). Linear operators. Matrix of a linear operator in general and orthogonal basis. Kernel and image. Composition of linear operators. Multiplication of matrices as a composition of operators. Change of basis and the matrix of linear operator. Inversion of a linear operator. Inversed matrix. Orthogonal operators and matrices.

 

Eigenvectors and eigenvalues. Characteristic polynomial. Calculation of eigenvalues and eigenvectors. Diagonalization of a matrix.

 

Definition of Hermitian operators via inner product. Eigenvalues and eigenvectors of Hermitian operators. Matrix of Hermitian operator in the orthogonal basis. Basis built from the eigenvectors of a Hermitian operator. Diagonalization of Hermitian operators. Examples of Hermitian operators in quantum mechanics.

 

Introduction to bilinear forms. Quadratic forms and inner product. Law of inertia. Hermitian forms.

 

 

ג. חובות הקורס:

 

דרישות קדם:  מתמטיקה 5 יחידות

 

חובות / דרישות / מטלות: תרגילי בית ובחינה

 

מרכיבי הציון הסופי (ציון מספרי / ציון עובר): הרצאה 80%, תרגיל 20%

 

 

ד. ביבליוגרפיה: (חובה/רשות)

    

     ספרי הלימוד (textbooks) וספרי עזר נוספים:

 

Outline) 1991 or later edition Lipschutz – Linear algebra (Schaum (חובה)

Axler - Linear Algebra Done Right, 2ed, 1997