Dimension: # of vectors in basis
dim Col A – number of pivot cols of A
dim Nul A - # free variables in A
(or number of non pivot cols of A)
Note: dim Col A + dim Nul A = n
DEFINITION: The set of all linear
combinations of the row vectors of a
matrix A, denoted by Row A
3
6 
 1 2
A   2  5  6  12
 1  3  3  6 
Row A = span {r1, r2, r3}
Where
r1=(-1, 2, 3, 6)
r2=(2, -5, -6, -12)
r3=(1, -3, -3, -6)
Note: Row A = Col AT
When we use row operations to
reduce Matrix A to Matrix B we are
taking linear combinations of the rows
of A to come up with B.
Recall Row A = span {r1, r2, r3} = all
linear combinations of the rows of A
This leads to the following Theorem
If two matrices A and B are row
equivalent then,
1. Row A = Row B
2. If B in echelon form, the nonzero
rows of B form a basis for Row A &
Row B
3
6 
  1 2 3 6
 1 2
A   2  5  6  12 B   0  1 0 0


 1  3  3  6 
 0 0 0 0
are row equivalent. Find a basis for
row space, column space and null
space of A. Also state the
dimension of each.
Very similar to the test
question for this section
DEFINITION: All are equivalent
• dim Col A
• # of pivot columns of A
• dim Row A
• # nonzero rows in reduced matrix of A
• Rank A
Since
• Row A = Col AT
We Have
• dim Row A = dim Col AT
• Rank A = Rank AT
For Amxn
Rank A + dim Nul A = n
OR
# of pivot columns + # of non pivot
columns = n
For Amxn
Rank A is as large as possible
OR
Rank = min(m,n)
Given A5x7 with 5 pivot columns, is A of
full rank?
dim Col A = Rank A=5 = min(5,7)
A is of Full Rank
Pivot in each row
For Amxn and m≤n
(see 4.6 #26 for m>n)
Full Rank
Pivot in each row
Columns of A Span RM
There is a solution for all b in Ax=b
Important in Statistics
EXAMPLE: Suppose that a 5 x 8 matrix A
has rank 5. Find dim Nul A, dim Row A and
rank AT. Is Col A= R5?
EXAMPLE: For a 9 x 12 matrix A, find
the smallest possible value of dim Nul A.
The Rank Theorem provides us with a powerful tool for
determining information about a system of equations.
EXAMPLE: A scientist solves a nonhomogeneous system of
50 equations in 54 variables and finds that exactly 4 of the
unknowns are free variables. Can the scientist be certain
that any associated nonhomogeneous system (with the
same coefficients) has a solution?
Very similar to the test
question for this section
For Amxn
• The columns of A form a basis of Rn
• Col A = Rn
• dim Col A = n
• Rank A = n
• Nul A = {0}
• dim Nul A = 0