Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Bases in R³

Description

This lecture covers the concept of bases and vector subspaces in R³, discussing how a set of vectors forms a base if it is impossible to express one vector as a combination of the others. It explains the properties of bases, such as being linearly independent and generating the entire space. The lecture also explores examples of bases in R³, including canonical bases and non-generating bases. The instructor demonstrates how to determine if a set of vectors forms a base by checking for linear independence and spanning properties.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K.

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.

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

Linear independence

In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

Basis (linear algebra)

In mathematics, a set B of vectors in a vector space V is called a basis (: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called . Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.

Related lectures (35)

Linear Independence and Basis

Explains linear independence, basis, and matrix rank with examples and exercises.

Linear Algebra: Vector Spaces

Explores vector spaces, subspaces, bases, and linear combinations in R² and R³, including free and linked families.

Linear Dependence and Independence: Properties and Criteria MOOC: Linear Algebra (Part 1)

Covers the properties and criteria of linear dependence and independence in a vector space.

Linear Algebra: Subspaces and Transformations

Explores subspaces in linear algebra and transformations, including kernels and images of linear transformations.

Linear Independence and Bases in Vector Spaces Log in to Mediaspace to watch this video