Section 4.6 (Rank)

1. Section 4.6 (Rank)

2. 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

3. Row Space DEFINITION: The set of all linear combinations of the row vectors of a matrix A, denoted by Row A

4. Example 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

5. 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

6. 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

7. 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

8. 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

9. Rank Property Since Row A = Col AT We Have dim Row A = dim Col AT Rank A = Rank AT

10. Rank Theorem For Amxn Rank A + dim Nul A = n OR # of pivot columns + # of non pivot columns = n

11. Full Rank For Amxn Rank A is as large as possible OR Rank = min(m,n)

12. 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

13. 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

14. 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.

15. 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 nonhomogeneoussystem 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 nonhomogeneoussystem (with the same coefficients) has a solution? Very similar to the test question for this section

16. 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