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Transcript of Dimension: # of vectors in basis dim Col A – number of pivot cols of A dim Nul A - # free...
Section 4.6 (Rank)
Recap Dimension
Dimension: # of vectors in basisdim Col A – number of pivot cols of Adim Nul A - # free variables in A
(or number of non pivot cols of A)Note: dim Col A + dim Nul A = n
Row Space
DEFINITION: The set of all linear combinations of the row vectors of a matrix A, denoted by Row A
Example
Row A = span {r1, r2, r3}
Wherer1=(-1, 2, 3, 6)r2=(2, -5, -6, -12)r3=(1, -3, -3, -6)
Note: Row A = Col AT
6331
12652
6321
A
Row A Observations
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
Theorem
If two matrices A and B are row equivalent then,1. Row A = Row B2. If B in echelon form, the nonzero
rows of B form a basis for Row A & Row B
Example
are row equivalent. Find a basis for row space, column space and null space of A. Also state the dimension of each.
6331
12652
6321
A
0000
0010
6321
B
Very similar to the test question for this section
Rank
DEFINITION: All are equivalent• dim Col A • # of pivot columns of A• dim Row A• # nonzero rows in reduced matrix of A• Rank A
Rank Property
Since • Row A = Col AT
We Have • dim Row A = dim Col AT
• Rank A = Rank AT
Rank Theorem
For Amxn
Rank A + dim Nul A = n
OR
# of pivot columns + # of non pivot columns = n
Full Rank
For Amxn
Rank A is as large as possible
OR
Rank = min(m,n)
Full Rank Example
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
Full Rank Observation
For Amxn and m≤n (see 4.6 #26 for m>n)
Full Rank Pivot in each rowColumns of A Span RM
There is a solution for all b in Ax=b
Important in Statistics
Section 4.6 ExamplesEXAMPLE: 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.
Section 4.6 ExampleThe Rank Theorem provides us with a powerful tool fordetermining 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
Additions to IMT
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