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