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

Lecture

Linear Combinations: Vectors and Matrices

In course

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications. ...

Description

This lecture introduces the concept of linear combinations of vectors and matrices, exploring how different vectors can be combined to generate new vectors in Rn. The instructor explains the operations of addition, scalar multiplication, and subtraction on vectors, highlighting the properties and geometric interpretations of these operations. Through examples, the lecture demonstrates how linear combinations of vectors can form lines, planes, or grids in Rn, depending on the number of vectors involved. The lecture also covers the use of matrices to represent linear transformations and how matrix-vector multiplication can be interpreted as a combination of the columns of the matrix. The discussion extends to solving systems of linear equations using matrix operations and understanding the span of vectors in Rn.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Nicolas Boumal

Jérôme Scherer

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

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 span

In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is defined as the set of all linear combinations of the vectors in S. For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain S, or as the smallest subspace containing S. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.

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.

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

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.

Related lectures (44)

Linear Independence and Basis

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

Linear Algebra: Matrix Operations and Basis

Explores matrix operations, rank determination, kernel dimensions, and basis concepts in linear algebra.

Matrix Operations: Linear Systems and Solutions Log in to Mediaspace to watch this video

Explores matrix operations, linear systems, solutions, and the span of vectors in linear algebra.

Linear Equations: Vectors and Matrices Log in to Mediaspace to watch this video

Covers linear equations, vectors, and matrices, exploring their fundamental concepts and applications.

Linear Transformations: Matrices and Applications Log in to Mediaspace to watch this video