what is vector space subspaces and vectors in linear algebra

In linear algebra, the concepts of vector spaces, subspaces, and vectors are foundational. Here's an overview of each concept: ### Vector Spaces A **vector space** (also called a linear space) is a collection of objects called **vectors**, which can be added together and multiplied by scalars. A vector space over a field \( F \) satisfies the following properties: 1. **Closure under Addition**: If \( \mathbf{u} \) and \( \mathbf{v} \) are vectors in the vector space \( V \), then \( \mathbf{u} + \mathbf{v} \) is also in \( V \). 2. **Closure under Scalar Multiplication**: If \( \mathbf{u} \) is a vector in \( V \) and \( c \) is a scalar in \( F \), then \( c\mathbf{u} \) is also in \( V \). 3. **Associativity of Addition**: For all \( \mathbf{u}, \mathbf{v}, \mathbf{w} \in V \), \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \). 4. **Commutativity of Addition**: For all \( \mathbf{u}, \mathbf{v} \in V \), \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \). 5. **Identity Element of Addition**: There exists a vector \( \mathbf{0} \) in \( V \) such that \( \mathbf{u} + \mathbf{0} = \mathbf{u} \) for all \( \mathbf{u} \in V \). 6. **Inverse Elements of Addition**: For every \( \mathbf{u} \in V \), there exists a vector \( -\mathbf{u} \in V \) such that \( \mathbf{u} + (-\mathbf{u}) = \mathbf{0} \). 7. **Distributive Properties**: - \( c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v} \) - \( (c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u} \) 8. **Associativity of Scalar Multiplication**: For all \( c, d \in F \) and \( \mathbf{u} \in V \), \( c(d\mathbf{u}) = (cd)\mathbf{u} \). 9. **Identity Element of Scalar Multiplication**: For all \( \mathbf{u} \in V \), \( 1\mathbf{u} = \mathbf{u} \). ### Subspaces A **subspace** is a subset of a vector space that is itself a vector space under the same operations (vector addition and scalar multiplication). For a subset \( W \) of a vector space \( V \) to be a subspace, it must satisfy the following conditions: 1. **Non-empty**: The zero vector \( \mathbf{0} \) must be in \( W \). 2. **Closed under Addition**: If \( \mathbf{u} \) and \( \mathbf{v} \) are in \( W \), then \( \mathbf{u} + \mathbf{v} \) must also be in \( W \). 3. **Closed under Scalar Multiplication**: If \( \mathbf{u} \) is in \( W \) and \( c \) is a scalar in \( F \), then \( c\mathbf{u} \) must also be in \( W \). ### Vectors **Vectors** are the elements of vector spaces. They can represent points in space, directions, or more abstract entities. Vectors can be represented in various forms, such as: - **Column vectors**: e.g., \(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\) for a 3-dimensional vector. - **Row vectors**: e.g., \(\begin{pmatrix} x & y & z \end{pmatrix}\). - **Polynomials**, **functions**, or **matrices**, depending on the specific vector space. Vectors are typically manipulated through operations like addition and scalar multiplication, adhering to the rules and properties outlined for vector spaces. ### Summary - A **vector space** is a set of vectors with specific operations. - A **subspace** is a subset of a vector space that is itself a vector space. - **Vectors** are the individual elements within these spaces, subject to linear operations. Understanding these concepts is crucial for further studies in linear algebra and its applications in various fields.