Lecture

Linear Algebra in R³: Vector Spaces

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Description

This lecture covers the concept of vector spaces in three dimensions, exploring the definition of a family of vectors, linear combinations, and subspaces in R³. The instructor explains how to determine if a family of vectors is free or linked, and how to find the coordinates of a vector in a given basis. The lecture also delves into the properties of subspaces in R³, including the conditions for a set to be a subspace and the alternatives when a vector is part of a subspace. Examples and exercises are provided to illustrate the theoretical concepts.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

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Real coordinate space

In mathematics, the real coordinate space of dimension n, denoted Rn or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. Special cases are called the real line R1 and the real coordinate plane R2. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors. The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space.

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

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