This preview shows pages 1–2. Sign up to view the full content.
Math 2331
Linear Algebra
Section 3.6
The Fundamental Subspaces of a Matrix
Matrix A is M rows by N columns
The fundamental subspaces associated with A are 
1. The column space of A C(A) or Col (A) all linear combinations of the column vectors
of A.
2. The null space of A  all solutions to the equation Ax = 0 and denoted N(A)
3. The Row space of A  all linear combinations of the row vectors of A.
ALSO  this is the COLUMN SPACE of the transpose of A.
(
)
T
CA
4. The left nullspace of A  The NULL SPACE of the transpose of A 
(
)
T
NA
Let's look at these one at a time, both for a rowreduced echelon form matrix and for a
general M x N matrix.
COLUMN SPACE
Remember that A has m rows and n columns, so we can think of A as n column vectors.
HOWEVER, How many elements are in each column vector?
So, the column space lives in Mdimensional space (the number of rows!!!!!!)
The column space is the span of all the column vectors. All of them. However, some of
the columns already may be linear combinations of the other columns so one of the
questions to ask is which ones do you really need to generate the column space. The
answer is that you only need the pivot columns. The other columns are combos of the
pivot columns.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Linear Algebra, Algebra, Vectors

Click to edit the document details