january 22
DESCRIPTION
January 22. Review questions. Math 307 Spring 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-Mail: [email protected] Office Hours: 9:00-10:00 MWF 2:00- 3:00 MWF. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/1.jpg)
January 22
Review questions
![Page 2: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/2.jpg)
• Math 307• Spring 2003• Hentzel• Time: 1:10-2:00 MWF• Room: 1324 Howe Hall• Instructor: Irvin Roy Hentzel• Office 432 Carver• Phone 515-294-8141• E-Mail: [email protected]• Office Hours: 9:00-10:00 MWF• 2:00- 3:00 MWF
![Page 3: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/3.jpg)
We apply the ideas we have learned
• We go over the 36 true false questions so that we learn how to use the facts we have learned so far to get really nice results.
![Page 4: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/4.jpg)
1 The following matrix is in RREF. | 1 2 0 | | 0 0 1 | | 0 0 0 |
True. | |_1__ 2_ 0 | | 0 0 |_ 1 _| | 0 0 0 | Find the: (a) Stairs (b) Stair step ones (c) Zeros below the stairs (d) Zeros above the stair step ones.
![Page 5: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/5.jpg)
2. A system of four equations in three unknowns is always inconsistent.
False: x + y + z = 3 x = 1 y = 1 z = 1Is consistent.
![Page 6: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/6.jpg)
3. There is a 3x4 matrix with rank 4.
False: A matrix with 3 rows can have at most three stair step ones. Thus the matrix can have rank at most 3.
![Page 7: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/7.jpg)
4. If A is a 3x4 matrix and vector V is in R^4, then vector AV is in R^3.
True: A V = (AV) 3x4 4x1 3x1
![Page 8: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/8.jpg)
5. If the 4x4 matrix A has rank 4, then any linear system with coefficient matrix A will have a unique solution.
True: The Row Canonical Form of [A | B] will always reduce to [ I | B* ] for some
B*. There is no stair step one in the last column so there is an answer. There are no parameters so the answer is unique.
![Page 9: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/9.jpg)
6. There exists a system of three linear equations with three unknowns that has exactly three solutions.
False: If there is more than one solution, then there is at least one parameter and there will be an infinite number of solutions.
![Page 10: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/10.jpg)
7. There is a 5x5 matrix A of rank 4 such that the system AX = 0 has only the solution X = 0.
False: Rank 4 means four stair step ones. One of the variables will not be above a stair step one. Therefore, there is at least one parameter. There may be no solutions at all, but if there are any solutions at, there are infinitely many of them.
![Page 11: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/11.jpg)
8. If matrix A is in RREF, then at least one of the entries in each column must be 1.
False: Consider this matrix in RREF. | |_1_ 0 0 0 | | 0 |_1_ 0_ 0 | | 0 0 0 |_1_|
![Page 12: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/12.jpg)
9. If A is an nxn matrix and X is a vector in R^n, then the product AX is a linear combination of the columns of the matrix A.
True: The columns of AB are linear combinations of the columns of A.
![Page 13: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/13.jpg)
10. If vector U is a linear combination of vectors V and W, then we can write U = aV+bW for some scalars a and b.
True: This is exactly what we mean when we say that a vector is a linear combination of two vectors.
![Page 14: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/14.jpg)
11. The rank of the following matrix is 2. | 2 2 2 | | 2 2 2 | | 2 2 2 |
False: Reduce it to Row Canonical Form and count the non zero rows. The rank is 1.
| 1 1 1 | | 0 0 0 | | 0 0 0 |
![Page 15: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/15.jpg)
12. | 11 13 15| | -1 | | 13 | | 17 19 21| | 3 | = | 19 | | -1 | | 21 |
False. It cannot possibly be correct sincea 2x3 matrix times a 3x1 matrix will be a2x1 matrix, not a 3x1 matrix.
![Page 16: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/16.jpg)
13. There is a matrix A such that
| -1 | | 3 | A | 2 | = | 5 | . | 7 |
True: | 0 3/2 | | 0 5/2 | | 0 7/2 |Is such a matrix.
![Page 17: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/17.jpg)
14. | 1 | Vector | 2 | is a linear combination of | 3 | | 4 | | 7 |Vectors | 5 | and | 8 | . | 6 | | 9 |
True: | 4 | | 7 | | 1 | 2 | 5 | - 1 | 8 | = | 2 | | 6 | | 9 | | 3 |
![Page 18: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/18.jpg)
15. The system below is inconsistent.
| 1 2 3 | | x | | 1 | | 4 5 6 | | y | = | 2 | | 0 0 0 | | z | | 3 |
True: The last equation requires that0x + 0 y + 0 z = 3 which cannot possiblebe true.
![Page 19: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/19.jpg)
16.There exists a 2x2 matrix A such that
A | 1 | = | 3 |. | 2 | | 4 |
True: | 1 1 | | 1 | = | 3 | | 2 1 | | 2 | | 4 |
![Page 20: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/20.jpg)
17. If A is a nonzero matrix of the form | a -b | | b a |Then the rank of A must be 2.
True: If a = 0, then the RCF = I and the rank is 2. If a =/= 0, then
| a -b | ~ | 1 -b/a | ~ | 1 -b || b a | | b a | | 0 (b^2)/a +a |
If (b^2)/a + a = 0, the rank is 1. If (b^2)/a =/= 0, the rank is 2.Since (b^2)/a + a = 0 requires a^2 + b^2 = 0 which cannothappen for real numbers a and b, we know that the rank isalways 2.
![Page 21: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/21.jpg)
18. The rank of this matrix is 3. | 1 1 1 | | 1 2 3 | | 1 3 6 |
True:| 1 1 1 | | 1 1 1 | | 1 0 -1 | | 1 0 0 || 1 2 3 | ~ | 0 1 2 | ~ | 0 1 2 | ~ | 0 1 0 || 1 3 6 | | 0 2 5 | | 0 0 1 | | 0 0 1 |
![Page 22: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/22.jpg)
19. The system is inconsistent for any (4x3) matrix A. | 0 | A X = | 0 | | 0 | | 1 |
False: | 0 0 0 | | 1 | | 0 | | 0 0 0 | | 1 | | 0 | | 0 0 0 | | 1 | = | 0 | | 1 0 0 | | 1 |
![Page 23: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/23.jpg)
20.There exists a 2x2 matrix A such that A | 1 | = | 1 | and A | 2 | = | 2 |
| 1 | | 2 | | 2 | | 1 |
False: A| 2 | = 2 A| 1 | = 2 | 1 | = | 2| =/= | 2 |
| 2 | | 1 | | 2 | | 4| | 1 |
![Page 24: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/24.jpg)
21. There exist scalars a and b such that this matrix has rank 3.
| 0 1 a || -1 0 b || -a -b 0 |
False: | 0 1 a | | -1 0 b | | 1 0 -b | | 1 0 –b | | 1 0 –b |
| -1 0 b |~| 0 1 a |~| 0 1 a |~| 0 1 a |~| 0 1 a ||-a -b 0 | | -a –b 0 | |-a –b 0 | | 0 –b –ab | | 0 0 0 |
The Row Canonical Form has exactly 2 non zero rows.
![Page 25: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/25.jpg)
22. If V and W are vectors in R^4, then V must be a linear combination of V and W.
True. V = 1 V + 0 W.
![Page 26: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/26.jpg)
23. If U, V, and W are nonzero vectors in R^2, then W must be a linear combination of U and V.
False. U = | 1 | V = | 1 | W = | 0 | | 0 | | 0 | | 1 |
![Page 27: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/27.jpg)
24. If V and W are vectors in R^4, then the zero vector in R^4 must be a linear combination of V and W.
True: 0 V + 0 W = 0.
![Page 28: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/28.jpg)
25. If A and B are any two 3x3 matrices of rank 2, then A can be transformed into B by means of elementary row operations.
False. Since Row Canonical Forms are unique, we simply display two rank two 3x3 matrices in Row Canonical Form.
| 1 0 0 | | 0 1 0 | | 0 1 0 | | 0 0 1 | | 0 0 0 | | 0 0 0 |
![Page 29: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/29.jpg)
26. If vector U is a linear combination of vectors V and W, and V is a linear combination of vectors P, Q and R, then U is a linear combination of P, Q, R, and W.
True: U = aV + b W and V = cP+dQ+eR, then U = a(cP+dQ+eR)+bW = acP+adQ+aeR+bW.
![Page 30: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/30.jpg)
27.A system with fewer unknowns than equations must have infinitely many solutions or none.
False: x = 1 has exactly one solution y = 1 x+y = 2
![Page 31: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/31.jpg)
28. The rank of any upper triangular matrix is the number of non zeros on the
diagonal.
False: | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | | 0 0 0 0 |
This matrix has rank 3 and no non zeros on the diagonal.
![Page 32: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/32.jpg)
29.If the system AX = B has a unique solution then A must be a square matrix.
False. | 1 0 0 | | x | | 1 | | 0 1 0 | | y | = | 1 | | 0 0 1 | | z | | 1 | | 1 1 1 | | 3 |
This has a unique solution and A is not square.
![Page 33: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/33.jpg)
30.If A is a 4x3 matrix, then there exists a vector B in R^4
such that the system AX = B is inconsistent.
True. The system
| 1 0 0 | 0 | | 0 1 0 | 0 | | 0 0 1 | 0 | | 0 0 0 | 1 |
is inconsistent. If we use inverse elementary row
operations, we can backwards transform this system to
any system [ A | B ] where A is a 4x3 matrix with rank 3.
The backwards transformed system will still be
inconsistent.
![Page 34: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/34.jpg)
31.If A is a 4x3 matrix of rank 3 and AV = AW for two vectors V and W in R^3, then the vectors V and W must be equal.
True: If A V = A W, then A(V-W) = 0. Since A has rank three,
the only solution to AX = 0 is X = 0. Thus V-W = 0 and V = W..
![Page 35: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/35.jpg)
32. If A is a 4x4 matrix and the system
| 2 |A X = | 3 | has a unique solution, | 4 | | 5 |then the system AX = 0 has only the solutionX = 0.
True. If any system has exactly one solution, thenthere will be no parameters. Thus every solution isunique.
![Page 36: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/36.jpg)
33. If vector U is a linear combination of vectors V and W, then W must be a linear combination of U and V.
False: | 1 | | 0 | | 2 | 2 | 0 | + 0 | 1 | = | 0 | | 0 | | 0 | | 0 |
v w u
2V+ 0 W = U, but W is not a linear combination of U and V
.
![Page 37: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/37.jpg)
| 1 0 2 |34 If A = [ U V W ] and RREF(A) = | 0 1 3 | | 0 0 0 |
then the equation W = 2U + 3 V must hold..
| -2 |True. The vector | -3 | must be in the null space of A | 1 |
So -2 U -3 V + W = 0. Thus W = 2 U + 3 V.
![Page 38: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/38.jpg)
35.If A and B are matrices of the same size, then the formula rank(A+B) = rank(A) + rank(B) must hold.
False
| 1 0 0 | | 0 1 0 | = | 1 1 0 | | 0 1 0 | + | 0 0 1 | | 0 1 1 |
The rank of all three of these matrices is two.
![Page 39: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/39.jpg)
36.If A and B are any two nxn matrices of rank n, then A can be transformed into B by elementary row operations.
True. Since A and B are both equivalent to the identity matrix I, we can transform either one of them into the other.
![Page 40: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/40.jpg)
![Page 41: January 22](https://reader035.vdocument.in/reader035/viewer/2022062816/5681510b550346895dbf27a6/html5/thumbnails/41.jpg)