Math 307

1

• Math 307
• Spring, 2003
• Hentzel
• Time 110-200 MWF
• Room 1324 Howe Hall
• Instructor Irvin Roy Hentzel
• Office 432 Carver
• Phone [phone]
• E-mail hentzel_at_iastate.edu
• http//www.math.iastate.edu/hentzel/class.307.ICN
• Text Linear Algebra With Applications, Second
  Edition

2

• Practice Test Test I,
• Actual test is Friday, February 12, 2003
• Hentzel
• 1. Solve AX B and write the answer as
• X X 0 a1 X 1 a2 X 2 ... ar
  Xr
• and check your answer using
• AX0 X1 X2 ... Xr B 0 0 0 ... 0.
• 1 2 0 1 2 3 x
  2
• 1 3 0 0 1 2 y
  1
• 2 5 0 1 3 5 z
  3
• 4 10 0 2 6 11 w
  7
• u
  
• v
  

3

• 1 2 0 1 2 3 x 2
• 1 3 0 0 1 2 y 1
• 2 5 0 1 3 5 z 3
• 4 10 0 2 6 11 w 7
• u
• v

4

• 1 2 0 1 2 3 2
• 1 3 0 0 1 2 1
• 2 5 0 1 3 5 3
• 4 10 0 2 6 11 7

5

• 1 0 0 3 4 0 -1
• 0 1 0 -1 -1 0 0
• 0 0 0 0 0 1 1
• 0 0 0 0 0 0 0

6

• 2. Find the inverse of this matrix.
• 1 1 2 0
• 1 2 3 0
• 2 0 5 1
• 2 3 4 0

7

• 1 1 2 0 1 0 0 0
• 1 2 3 0 0 1 0 0
• 2 0 5 1 0 0 1 0
• 2 3 4 0 0 0 0 1

8

• 1 0 0 0 1 -2 0 1
• 0 1 0 0 -2 0 0 1
• 0 0 1 0 1 1 0 -1
• 0 0 0 1 -7 -1 1 3

9

• 3. (a) Write the matrix of the linear
  transformation of differentiation with respect to
  the basis
• e2x , x e2x , x 2 e2x , x3 e 2x .
• (b) Find some way to use the matrix from part
  (a) to compute the eighth derivative of
• 2 3x 5x2 - x3 e2x .

10

• 2 1 0 0 2 7 30 106 320 864 2144
  4960
• 0 2 2 0 3 16 46 108 224 416 672
  832
• 0 0 2 3 5 7 8 4 -16 -80
  -256 -704
• 0 0 0 2 -1 -2 -4 -8 -16 -32
  -64 -128
• 10752
• 256
• -1792
• -256

11

• 4. Multiply these two matrices.
• 1 0 0 0 1 0 3 8 2 4
• 0 1 0 0 0-1 1 0 1 2
• 0 0 2 0 0 0 3 3 1 2
• 1 1 1 0 0 0 4 3 1 0
• 0 0 0 0 1 1 5 4 3 2
• 1 3 9 2

12

• 5. Tidbits
• (a) The rows of AB are linear combinations of
  what?

13

• (b) What is the rank of a matrix?

14

• (c) What is the relation between the rank, the
  nullity, and the number of columns of a matrix?

15

• (d) What is the relationship between the Row
  Canonical Form of a matrix and the existence of
  the inverse of the matrix?

16

• (e) What is the 2x2 matrix which rotates the
  plane through 60 degrees?

17

• (f) Give a non zero 2x2 matrix which is not
  invertible.

18

• (g) What are the three elementary row
  operations.

19

• (h) When is a function a linear transformation?

20

• (j) Give three 2x2 matrices A, B, C such that
  (AB)C / A(BC)