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Semester 6: B.Sc. Mathematics
Vector spaces and subspaces
Vector spaces and subspaces
Definition of Vector Spaces
Vector spaces are mathematical structures formed by a collection of vectors, which can be added together and multiplied by scalars. A vector space must satisfy certain axioms, including closure under addition and scalar multiplication, existence of an additive identity, and the existence of additive inverses.
Properties of Vector Spaces
Key properties of vector spaces include commutativity and associativity of vector addition, distributive properties of scalar multiplication, and the existence of a multiplicative identity. These properties establish the foundational rules for vector space operations.
Subspaces
A subspace is a subset of a vector space that is itself a vector space under the same operations. A subspace must include the zero vector, be closed under addition, and be closed under scalar multiplication.
Examples of Vector Spaces
Common examples include the space of all n-tuples of real numbers, the space of all polynomials of a certain degree, and function spaces where functions are viewed as vectors.
Applications of Vector Spaces
Vector spaces are widely used in different fields, including physics for state spaces, computer science for graphics, and data science for modeling.
Basis and Dimension
A basis of a vector space is a set of linearly independent vectors that span the space. The dimension of the vector space is the number of vectors in the basis.
Linear Independence
A set of vectors is linearly independent if no vector can be written as a linear combination of the others. This concept is crucial in determining the basis and dimension of vector spaces.
Linear independence, basis and dimension
Linear Independence, Basis, and Dimension
Linear Independence
A set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the others. Mathematically, for vectors v1, v2, ..., vn in vector space V, they are linearly independent if the equation a1*v1 + a2*v2 + ... + an*vn = 0 has only the trivial solution (a1 = a2 = ... = an = 0).
Span
The span of a set of vectors is the set of all possible linear combinations of those vectors. If a set of vectors spans a vector space V, it means every vector in V can be expressed as a linear combination of the vectors in the set.
Basis
A basis of a vector space V is a set of linearly independent vectors that spans the space V. The number of vectors in a basis is called the dimension of the vector space. Every vector in V can be uniquely expressed as a linear combination of the basis vectors.
Dimension
The dimension of a vector space is defined as the number of vectors in a basis for that space. It is a measure of the 'size' or 'degrees of freedom' in the space. For example, the dimension of R^n is n, meaning it is spanned by n linearly independent vectors.
Linear transformations and matrix representation
Linear transformations and matrix representation
Definition of Linear Transformations
A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. Formally, if T is a linear transformation, then for vectors u and v, and scalar c, the following holds: T(u + v) = T(u) + T(v) and T(cu) = cT(u).
Properties of Linear Transformations
Linear transformations have certain properties, including linearity, the mapping of zero vector to zero vector, and the preservation of linear combinations. They can be represented in a systematic way using matrices.
Matrix Representation of Linear Transformations
Every linear transformation can be represented by a matrix. If T is a linear transformation from R^n to R^m, then there exists a matrix A such that for any vector x in R^n, T(x) = Ax. The matrix A encodes the action of T.
Dimension and Rank of Linear Transformations
The rank of a linear transformation is the dimension of its image, while the nullity is the dimension of its kernel. The rank-nullity theorem states that for a linear transformation from R^n to R^m, rank(T) + nullity(T) = n.
Applications of Linear Transformations
Linear transformations are widely used in various fields including computer graphics, engineering, and data science. They are essential in transforming geometric shapes and performing operations like rotations and scaling.
Eigenvalues, eigenvectors, diagonalization
Eigenvalues, Eigenvectors, Diagonalization
Definition and Properties
Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvalue is a scalar λ such that there exists a non-zero vector v (the eigenvector) satisfying the equation Av = λv. Key properties include: 1. The eigenvalues of a matrix are the roots of its characteristic polynomial. 2. The eigenvectors corresponding to a distinct eigenvalue are linearly independent.
Finding Eigenvalues and Eigenvectors
To find eigenvalues of a matrix A, calculate the characteristic polynomial given by det(A - λI) = 0, where I is the identity matrix. Solving this polynomial gives the eigenvalues. For each eigenvalue λ, find the corresponding eigenvectors by solving the equation (A - λI)v = 0.
Diagonalization
A matrix A is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that A = PDP^-1. The columns of P are the eigenvectors of A, and the diagonal entries of D are the corresponding eigenvalues. A matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the dimension of A.
Applications of Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors have numerous applications in various fields, including physics, engineering, and machine learning. For example, they are used in solving systems of differential equations, in principal component analysis (PCA) for dimensionality reduction, and in stability analysis of dynamical systems.
Theorems Related to Eigenvalues and Eigenvectors
Several important theorems revolve around eigenvalues and eigenvectors. For instance, the spectral theorem states that every symmetric matrix can be diagonalized by an orthogonal matrix, implying that its eigenvalues are real and its eigenvectors are orthogonal. Another crucial theorem is the Gershgorin Circle Theorem, which provides bounds on the location of eigenvalues in the complex plane.
Inner product spaces and Gram-Schmidt orthogonalization
Inner product spaces and Gram-Schmidt orthogonalization
Definition of Inner Product Spaces
An inner product space is a vector space V equipped with an inner product (·,·) that associates a real number with every pair of vectors in V. This product must satisfy the following properties: positivity, linearity in the first argument, symmetry, and conjugate symmetry.
Examples of Inner Product Spaces
Common examples include Euclidean spaces where the inner product is the dot product. Function spaces can also be inner product spaces with inner products defined through integrals. For instance, for functions f and g defined on an interval, the inner product can be defined as ∫ f(x)g(x) dx.
Orthogonality in Inner Product Spaces
Vectors are said to be orthogonal if their inner product is zero. This concept extends to define orthogonal sets and orthonormal sets where each vector in the set has a unit norm. This is crucial for simplifying problems in linear algebra.
The Gram-Schmidt Process
The Gram-Schmidt procedure is a method for orthonormalizing a set of vectors in inner product spaces. Given a linearly independent set of vectors, the process produces an orthonormal set of vectors, maintaining their span. This is achieved by sequentially subtracting projections of previous vectors.
Applications of Gram-Schmidt Orthogonalization
Gram-Schmidt is used in various applications such as numerical methods, computer graphics, and quantum mechanics. It simplifies computations involving projections and helps in improving numerical stability in algorithms.
Advantages and Limitations of Gram-Schmidt
While the Gram-Schmidt process is simple and intuitive, it has numerical instability issues, and modified versions exist (like Modified Gram-Schmidt) to address these problems. Understanding stable algorithms is crucial for practical applications.
