LINEAR SUBSPACE Meaning and
Definition
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A linear subspace, also known as a vector subspace or simply a subspace, is a fundamental concept in linear algebra. It refers to a subset of a vector space that is closed under addition and scalar multiplication. In other words, a linear subspace is a collection of vectors that, when added together or multiplied by scalars, remain within that subspace.
To be more specific, a linear subspace V is defined as a subset of a vector space over a field F, satisfying the following conditions:
1. The zero vector is an element of V, denoted as 0 ∈ V. In other words, the subspace must contain the origin of the vector space.
2. The subspace is closed under vector addition. If u and v are vectors in V, then their sum u + v is also a vector in V.
3. The subspace is closed under scalar multiplication. If u ∈ V and c is a scalar in F, then the product cu is also a vector in V.
These conditions ensure that a linear subspace preserves the vector space structure, meaning it inherits the properties of the larger vector space. For example, if the original vector space is finite-dimensional, the linear subspace will also be finite-dimensional.
Linear subspaces are widely used in various branches of mathematics and physics. They form the building blocks for concepts like linear independence, span, and basis, providing a foundational framework for studying vector spaces and their applications.
Etymology of LINEAR SUBSPACE
The word "linear" is derived from the Latin word "linearis", which means "belonging to a line". It comes from the noun "linea", meaning "a line".
The term "subspace" is composed of two parts: "sub-" and "-space". The prefix "sub-" comes from the Latin word "sub", meaning "under" or "below". It indicates that the subspace is a subset or smaller component of a larger space. The suffix "-space" is derived from the Latin word "spatium", meaning "space" or "room". It refers to a mathematical concept of a set of elements with certain properties.
Therefore, the term "linear subspace" denotes a subset within a larger space that possesses specific linearity properties.
