Based on the following axioms of vector spaces where u & v are vectors and c is a scalar:
• u + v is in V
• c*u is in V
- I thought that the "space" of a specific matrix's vector space was all the linear combinations of its vectors, but I later found that to be the column space of that particular matrix.
If Null and Column spaces are subspaces of a Matrix's vector space, then what exactly makes up the Vector Space of a matrix? Is it the set of vectors as a whole?
Also, the definition of a subspace states that it is a "subset" of vectors from a larger vector space. The column Space seems to encompass all the vectors of a vector space while the null space has an entirely different set of vectors from the original vector space set.
The Null space in particular doesn't feel like a subset of a matrix's vector space, so I'm likely missing an idea of Vector- and Subspaces. What exactly constitutes a subset of a vector space?