math 307 spring, 2003 hentzel time: 1:10-2:00 mwf room: 1324 howe hall office 432 carver phone...
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Math 307Spring, 2003
Hentzel
Time: 1:10-2:00 MWFRoom: 1324 Howe Hall
Office 432 CarverPhone 515-294-8141
E-mail: [email protected]
http://www.math.iastate.edu/hentzel/class.307.ICN
Text: Linear Algebra With Applications, Second Edition Otto Bretscher
Friday, Feb 7 Chapter 2
No hand-in-homework assignment
Main Idea: I do not want any surprises on the test.
Key Words: Practice test
Goal: Test over the material taught in class.
1. The function T|x| = |x-y| is a
|y| |y-x|
linear transformation.
True. It has matrix | 1 -1 |.
|-1 1 |
• 3. If A is any invertible nxn matrix, then
• rref(A) = In.
• True. A matrix is invertible if and only
• if its RCF is the identity.
• 5. The formula AB=BA holds for all nxn
• matrices A and B.
• False. | 0 1| |0 0| =/= | 0 0 | | 0 1 |
• | 0 0| |1 0| | 1 0 | | 0 0 |
• 6. If AB = In for two nxn matrices A and B,
• then A must be the inverse of B.
• True. This is false if A and B are not
• square.
• 7. If A is a 3x4 matrix and B is a 4x5 matrix, then AB will be a 5x3 matrix.
• False. AB will be a 3x5 matrix.
• 8. The function T|x| = |y| is a linear
• |y| |1|
• transformation.
• False. T (2 |0|) = |0| =/= 2 T|0| = | 0 |
• |0| |1| |0| | 2 |
• 9. The matrix | 5 6 | represents a• |-6 5 |
• rotation-dilation.
• True. The dilation is by Sqrt[61] the angle
• is ArcTan[-6/5] = -0.876058 radians
• 10. If A is any invertible nxn matrix, then
• A commutes with A-1.
• True. By definition, A A-1 = A-1A = I
• | 1 1 1 |
• Matrix | 1 0 1 | is invertible.
• | 1 1 0 |
•
• True.
• | 1 1 1 | | 1 0 1| | 1 0 0 |
• | 1 0 1 | ~ | 0 1 0| ~ | 0 1 0 |
• | 1 1 0 | | 0 1 -1| | 0 0 1 |
• 13. There is an upper triangular 2x2
• matrix A such that A2 = | 1 1 |
• | 0 1 |
• True. A = | 1 1/2 | is one possibility.
• | 0 1 |
• 14. The function
• T|x| = |(y+1)2 – (y-1)2 | is a linear• |y| |(x-3)2 – (x+3)2 |• transformation.
• True. T|x| = | 4 y|.• |y| |-12 x|
• 15. Matrix | k -2 | is invertible for all
• | 5 k-6 |
• real numbers k.
• True.
• | k -2 | ~ | 1 (k-6)/5 | ~ | 1 (k-6)/5 |
• | 5 k-6 | | k -2 | | 0 (-k^2+6k-10)/5|
• This polynomial has roots 3 (+/-) i so for all REAL numbers k, the RCF is I and it is invertible.
• 16. There is a real number k such that the
• matrix | k-1 -2 | fails to be invertible.
• | -4 k-3 |
• True. k = -1 | -2 -2 | k = 5 | 4 -2 |.
• | -4 -4 | | -4 2 |
• 17. There is a real number k such that
• the matrix | k-2 3 | fails to be
• | -3 k-2 |
• invertible.
• False.
• | k-2 3 | ~ | 1 -(k-2)/3 | ~ | 1 -(k-2)/3 |
• | -3 k-2 | | k-2 3 | | 0 (k-2)2+3|
• the roots are k = 2 (+/-) i Sqrt[3] which are
• not real.
• 18. Matrix | -0.6 0.8 | represents a
• |-0.8 -0.6 |
• rotation.
• True: theta = Pi + ArcCos[0.6] = 4.06889
• 20. There is a matrix A such that
• | 1 2 | A | 5 6 | = | 1 1 |.• | 3 4 | | 7 8 | | 1 1 |
• True | 1 2 | -1 | 1 1 | | 5 6 ||-1
• | 3 4 | | 1 1 | | 7 8 |
• Should work. 1/2 | 1 -1 |• | -1 1 |
• 21. There is a matrix A such that
• A | 1 1 | = | 1 2 |.• | 1 1 | | 1 2 |
• False Any linear combination of the rows• of | 1 1 | will look like | x x |.• | 1 1 | | y y |
• 22. There is a matrix A such that
• | 1 2 | A = | 1 1 |,
• | 1 2 | | 1 1 |
• True. | 1 1 | works.
• | 0 0 |
• 23. Matrix | -1 2 | represents a shear.• | -2 3 |
• False • | -1 2 | |x| = | -x + 2y| = |x| +2(-x+y) | 1|• | -2 3 | |y| | -2x+3y| |y| | 1|
• The fixed vector has | 1 |.• | 1 |•
• 25. The matrix product
• | a b | | d -b | is always a scalar
• | c d | | -c a |
• of I2.
• True. The scalar is ad-bc.
• 26. There is a nonzero upper triangular
• 2x2 matrix A such that A2 = | 0 0 |.
• | 0 0 |
• True. A = | 0 1 | is one possibility.
• | 0 0 |
• 27. There is a positive integer n such that
• | 0 -1 | n = I2.
• | 1 0 |
• True. n = 4 is one possibility.
• 28. There is an invertible 2x2 matrix A
• such that A-1 = | 1 1 |.• | 1 1 |
• False. The RCF of | 1 1 | = | 1 1 |• | 1 1 | | 0 0 |
• so | 1 1 | cannot be an invertible matrix.• | 1 1 |
• 29. There is an invertible nxn matrix with two identical rows.
• False. If A has two identical rows, then
• AB has 2 identical rows also. Thus
• AB cannot be I.
• 31. If A17 = I2, then A must be I2.
• False A = | Cos[t] -Sin[t] |
• | Sin[t] Cos[t] |
• Where t = 2 Pi/17 should work.
• 32. If A2 = I2 , then A must be either I2 or –I2.
• False A = | -1 0 | is one possibility.
• | 0 1 |
• 33. If matrix A is invertible, then matrix
• 5 A is invertible as well.
• True. And (5A)-1 = 1/5 A-1.
• 34. If A and B are two 4x3 matrices such
• that AV = BV for all vectors v in R3, then
• matrices A and B must be equal.
• True. It follows that AI = BI for the 3x3
• identity matrix I. Thus A=B.
• 35. If matrices A and B commute, then the
• formula A2B = BA2 must hold.
• True. A2B = AAB = ABA=BAA=BA2.
• 36. If A2 = A for an invertible nxn matrix
• A, then A must be In.
• True. Multiply through by A-1 giving A=I.
• 37. If matrices A and B are both invertible,
• then matrix A+B must be invertible as well.
• False. Let B = -A.
• 38. The equation A2 = A holds for all 2x2
• matrices A representing an orthogonal
• projection.
• True. Once you have projected once by
• A, subequent actions by A will simply fix the
• vector.•
• 39. If matrix | a b c | is invertible, then• | d e f |• | g h I |
• matrix | a b | must be invertible as well.• | d e |
• | 0 0 1 |• False. | 0 1 0 | Is an example.• | 1 0 0 |•
• 40. If A2 is invertible, then • matrix A itself must be invertible.
• True. For A2 to be defined, then
• A must be square. If AAB = I, then
• A must be right invertible so A is• invertible.
• 41. The equation A-1 = A holds for all 2x2
• matrices A representing a reflection.
• True. For a reflection A2 = I.
• 42. The formula (AV).(AW) = V.W holds
• for all invertible 2x2 matrices A and for
• all vectors V and W in R2.
• False. | 1 1 | | 0 | .| 1 1 | | 1 | = 1• | 0 1 | | 1 | | 0 1| | 0 |
• 43. There exist a 2x3 matrix A and a 3x2
• matrix B such that AB = I3.
• True. | 1 0 0 | | 1 0 | = | 1 0 |
• | 0 1 0 | | 0 1| | 0 1 |
• | 0 0|
• 44. There exist a 3x2 matrix A and a 2x3
• matrix B such that AB = I3.
• False. There must be some X =/= 0
• such that BX = 0. Then 0 = ABX = X.
• Contradiction.
• 45. If A2 + 3A + 4 I3 = 0 for a 3x3 matrix
• A then A must be invertible.
• True. A(A+3) = -4 I3
• so the inverse of A is (-1/4)(A+3).
• 46. If A is an nxn such that A2 = 0, then
• matrix In+A must be invertible.
• True. (In+A)(In-A) = I.
• 47. If matrix A represents a shear, then• • the formula A2-2A+I2 = 0 must hold.
• True. (A-I)X will be a fixed vector.
• So A(A-I)X = (A-I)X which means
• A2-2A+I = 0.
• 48. If T is any linear transformation
• from R3 to R3, then T(VxW) = T(V)xT(W)
• for all vectors V and W in R3.• | 1 0 1 | | 1 | | 0 |• False. T = | 0 1 1 | V = | 0 | W = | 0 |• | 0 0 1 | | 0 | | 1 |• • | 0 | | 0 | | 1 | | 1 | | 0 |• T[VxW] = T| -1 | = |-1 | (TV)x(TW) = | 0 | x| 1 } = | -1 |.• | 0 | | 0 | | 0 | } 1 } | 1 |•
• 49. There is an invertible 10x10 matrix
• that has 92 ones among its entries.
• False. There are only 8 entries which
• are not one. At least 2 columns have
• only ones. Matrices with 2 identical
• columns are not invertible.
• 50. The formula rref(AB) = rref(A)rref(B)• holds for all mxn matrices A and for all• nxp matrices B.
• False A = B = | 0 0 |• | 1 0 |• rref(AB) =| 0 0 | rref(A)rref(B) = | 1 0 |• | 0 0 | | 0 0 |