OEF vector subspaces --- Introduction ---

This module gathers for the time being 8 exercises on subspaces of vector spaces.

Dimension of intersection

Fill-in: Let F be a vector space of dimension , and let , be two vector subspaces of F, of dimensions respectively and . Then is at least and at most .

Dim subspace by system

Let E be a sub-vector space of R defined by a homogeneous linear system. This system is composed of equations, and the rank of the coefficient matrix of this system is equal to . What is the dimension of E?

Dimension of sum

Fill-in: Let F be a vector space of dimension , and let , be two vector subspaces of F, of dimensions respectively and . Then is at least and at most .

Subbase

Fill-in: let F be a vector space of dimension , and let B be a basis of F. Let be a subset of de elements, and let E be the vector subspace of F generated by . Then dim(E) is equal to .

Subbase II

Fill-in: let F be a vector space of dimension , and let B be a basis of F. Let and be two subsets of B, with respectively and elements. Suppose that

has elements. Let and be the vector subspaces of F generated respectively by and , and let .

Then is equal to .

Dimension of subspace

Fill-in: Let E be a vector subspace of RR . Then dim(E) is equal to .

Dim subspace of matrices

Fill-in: let M× be the vector space over RR of × matrices, and let E be the vector subspace of M× consisting of matrices A such that =0, where B is a fixed non-zero matrix of dimension ×. Then dim(E) is at least , and at most .

Extension of subspace

Let F be a vector space of dimension , E a subspace of F generated by a set S, with . Let v be a vector of F which a linear combination of vectors in S, and let be the vector subspace of F generated by . What is the dimension of  ?
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