Dimension of intersection

Fill-in: Let $m_F be a vector space of dimension $val9, and let , be two vector subspaces of $m_F, of dimensions respectively $val10 and $val11. Then is at least and at most .

Dim subspace by system

Let E be a sub-vector space of R^$val9 defined by a homogeneous linear system. This system is composed of $val7 equations, and the rank of the coefficient matrix of this system is equal to $val6. What is the dimension of E?

Dimension of sum

Fill-in: Let $m_F be a vector space of dimension $val9, and let , be two vector subspaces of $m_F, of dimensions respectively $val10 and $val11. Then is at least and at most .

Subbase

Fill-in: let $m_F be a vector space of dimension $val9, and let $m_B be a basis of $m_F. Let be a subset of de $val10 elements, and let $m_E be the vector subspace of $m_F generated by . Then dim(E) is equal to .

Subbase II

Fill-in: let $m_F be a vector space of dimension $val9, and let $m_B be a basis of $m_F. Let and be two subsets of $m_B, with respectively $val10 and $val11 elements. Suppose that

has $val13 elements. Let and be the vector subspaces of $m_F generated respectively by and , and let .

Then is equal to .

Dimension of subspace

Fill-in: Let E be a vector subspace of $m_RR^$val11 $val15. Then dim(E) is equal to .

Dim subspace of matrices

Fill-in: let M_$val6×$val7 be the vector space over $m_RR of $val6×$val7 matrices, and let E be the vector subspace of M_$val6×$val7 consisting of matrices A such that $val11=0, where B is a fixed non-zero matrix of dimension $val8×$val9. Then dim(E) is at least , and at most .

Extension of subspace

Let $m_F be a vector space of dimension $val11, $m_E a subspace of $m_F generated by a set $m_S, with . Let $m_v be a vector of $m_F which $val9 a linear combination of vectors in $m_S, and let be the vector subspace of $m_F generated by . What is the dimension of ?