Null space and Column Space

quantumlight

I am just wondering what is meant when someone says the Col A is a subspace of null Space of A. What can you infer from that?

Also what is a null space of A(transpose)A

How do they relate to A? Are there theorems about this that I can look up?

HallsofIvy

First, if A maps vector space U to vector space V, the column space of A is a subset of and the null space of A is a subset of U so in order for that to makes sense U and V must be the same: A maps a space U to itself. In terms of matrices, that means A must be a square matrix. The columns space is the "range" of A. If y is in the column space of A, that means there exist some x such that Ax= y. If y is also in the null space, then Ay= A(Ax)= 0. Finally, if the column space is a subset of the null space, that must always be true: A(Ax)= A²x= 0. Again, in terms of matrices that means A²= 0.