§ Four fundamental subspaces
Let be . The Null space of is in . The column
space is in . The rows of are in . The nullspace
of is in .
We want a basis for each of those spaces, and what are their dimensions?
- Column space / Image: , since it corresponds to
- Null space .
- Row space: row spans the row space, so it's all linear combinations of the rows of . This is the same as all combinations of the columns of . Row space is denoted by .
- Null space of : , also called as the "left null-space of ".
- The dimension of the column space is the rank .
- The dimension of the row space is also the rank .
- The dimension of the nullspace is .
- Similarly, the left nullspace must be .
§ Basis for the column space
The basis is the pivot columns, and the rank is .
§ Basis for the row space
. Row operations do not preserve the column space, though
they have the same row space. The basis for the row space of and
since they both have the space row space, we just read off the first
rows of .
§ Basis for null space
The basis will be the special solutions. Lives in
§ Basis for left null space
It has vectors such that . We can equally write this as
. Can we infer what the basis for the left null space is
from the process that took us from to ? If we perform gauss-jordan,
so we compute the reduced row echelon form of ,
we're going to get where is whatever the identity matrix became.
Since the row reduction steps is equivalent to multiplying by some matrix ,
we must have that:
So the matrix that takes to is ! We can find the basis for the left
nullspace by lookinag at , because gives us .