Lecture

Rank Theorem: Matrix Dimension and Subspaces

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In course

MATH-111(f): Linear Algebra

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

Description

This lecture covers the rank theorem, which relates the dimensions of the kernel and image of a matrix. Starting with the characterization theorem, the instructor explains the rank of a matrix and its relationship with the dimensions of subspaces generated by the matrix's rows and columns. The lecture progresses to discuss equivalent matrices along lines, linearly independent vectors, and the conditions for invertible matrices. Demonstrations are provided to illustrate the concepts, leading to a detailed understanding of the rank theorem and its implications.

Instructor

Mikaël Cavallin

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

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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.

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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.

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