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Week of...

Notes and Links

1

Sep 11

About, Tue, HW1, Putnam, Thu

2

Sep 18

Tue, HW2, Thu

3

Sep 25

Tue, HW3, Photo, Thu

4

Oct 2

Tue, HW4, Thu

5

Oct 9

Tue, HW5, Thu

6

Oct 16

Why?, Iso, Tue, Thu

7

Oct 23

Term Test, Thu (double)

8

Oct 30

Tue, HW6, Thu

9

Nov 6

Tue, HW7, Thu

10

Nov 13

Tue, HW8, Thu

11

Nov 20

Tue, HW9, Thu

12

Nov 27

Tue, HW10, Thu

13

Dec 4

On the final, Tue, Thu

F

Dec 11

Final: Dec 13 25PM at BN3, Exam Forum

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Review of Last Class
Problem. Find the rank (the dimension of the image) of a linear transformation whose matrix representation is the matrix A shown on the right.

.

Theorem 1. If is a linear transformation and and are invertible linear transformations, then the rank of is the same as the rank of .


Proof. Owed.

Theorem 2. The following row/column operations can be applied to a matrix by multiplying it on the left/right (respectively) by certain invertible "elementary matrices":
 Swap two rows/columns
 Multiply a row/column by a nonzero scalar.
 Add a multiple of one row/column to another row/column.


Proof. Semiowed.

Solution of the problem. using these (invertible!) row/column operations we aim to bring to look as close as possible to an identity matrix, hoping it will be easy to determine the rank of the matrix we get at the end:
Do

Get

Do

Get

1. Bring a to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by .


2. Add times the first row to the third row, in order to cancel the in position 31.


3. Likewise add times the first row to the fourth row, in order to cancel the in position 41.


4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling).


5. Turn the 22 entry to a by multiplying the second row by .


6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" at position 22.


7. Using three column operations clean the second row except the pivot.


8. Clean up the row and the column of the in position 33 by first multiplying the third row by and then performing the appropriate row and column transformations. Notice that by pure luck, the at position 45 of the matrix gets killed in action.


But the matrix we now have represents a linear transformation satisfying for some bases of and of . Thus the image (range) of is spanned by , and as these are independent, they form a basis of the image. Thus the rank of is . Going backward through the "matrix reduction" process above and repeatedly using theorems 1 and 2, we find that the rank of must also be .
Class Notes
Scan of Week 9 Lecture 2 notes
Tutorial Notes
Nov09 Lecture notes 1 of 3
Nov09 Lecture notes 2 of 3
Nov09 Lecture notes 3 of 3
Scan of Week 9 Tutorial notes