 # Section 4.6 (Rank).

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Section 4.6 (Rank)

Recap Dimension 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

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} Where r1=(-1, 2, 3, 6)
Note: Row A = Col AT

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, Row A = Row B
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. 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 row Columns of A Span RM There is a solution for all b in Ax=b Important in Statistics

Section 4.6 Examples 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.

Section 4.6 Example 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

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