Matrix theory—linear subspace, generated subspace, kernel space, zero degree, intersection and sum of subspaces, direct sum

Linear subspace definition 

 If , V 1 is called a trivial subspace, otherwise it is called a non-trivial subspace.

generate subspace

 

Nuclear space, zero degree 

 untie:

rank(A)=2; n(A)=N-rank(A)=3-2=1, where N represents the number of unknown quantities.
n(A) can also be understood as the number of basic solution systems, that is, how many vectors there are in the basic solution system.

Conclusion:
(1) rnak(A) + n(A) = Number of columns of A
(2) n(A) - n(A^T) = (Number of columns of A) - (Number of rows of A)

Intersection and sum of subspaces

 example:

Naokazu 

Comprehensive examples:

 untie:

 Another example:

 

Another example: 

Then find the dimension of the intersection of the two subspaces: