Maple1231W5T5 - Further Linear Algebra I

Friday, 24 August 2018

1:04 PM

Machine generated alternative text:
Suppose that 
111 — 
609 
93 
398 
-537 
-330 
-316 
985 
-50 
-962 
507 
54 
-268 
-445 
788 
113 
-431 
-234 
439 
-785 
-718 
-215 
14 
and 
114 
-805 
-755 
20 
165 
214 
716 
851 
To avoid typing errors; you can copy and past the following sequences to your Maple worksheet to form entries of the vectors 
609, 93, 398, -537, -33€, -316, 985 
-5€, 
-431, 
-8€5, 
-962, 
-234, 
-755, 
5€7, 54, -268, -445, 788 
439, -785, -718, -215, 14 
2e, 165, 214, 716, 851 
(a) Find the dot product of and 112_ Enter your answer in the box below. 
1058112

 

Machine generated alternative text:
(609: 
( -431:-234.439: 14) 
with(LtnearÅ lgebra) 
DotProhct(u1: u2) 
1058112 
ul _u2 
1058112 
u2 ul 
1058112

 

Machine generated alternative text:
(b) Suppose that A is the matrix whose columns are these four vectors in order, 
4 
-46 
Let v be the vector 
16 
12 
Enter the values of Al, 
A = (ul 1 114) 
Hence Av is a linear combination of the form 
in the boxes below 
4 
0 
-46 
b7)T 
16 
12 
(c) Suppose that Av = (bl b2 
Enter the value of b6 in the box below 
b6 = 24358

 

Machine generated alternative text:
(ullu21u31u4) 
(4-46, 16, 12) 
Transpose( %) 
609 
93 
398 
-537 
-330 
-316 
985 
-962 
507 
54 
-268 
—445 
788 
-431 
-234 
439 
-785 
-718 
-215 
14 
- 805 
-755 
20 
165 
214 
716 
-11820 31820 
- 14466 
4 
-46 
16 
12 
-11820 
31820 
- 14466 
-15212 
2088 
24358 
-21872 
-15212 2088 24358 
-21872

 

Machine generated alternative text:
Suppose that S 
{111, 113, u4} where 
113 — 
205 
269 
264 
-157 
20 
171 
164 
263 
-232 
-62 
-54 
-229 
-212 
-221 
-260 
114 
-111 
146 
-281 
192 
-281 
4591 
18233 
11908 
6131 
584 
To avoid typing errors; you can copy and past the following sequences to your Maple worksheet to form entries of the vectors or an augmented matrix. 
2€5, 
171, 
-54, 
-111, 
4591, 
269, 264, -157, 20 
164, 263, -232, -62 
-229, -212, 
-221, -260 
146, -281, 192, 
-281 
18233, 11908, 6131, 584 
The vector v is in the span of S written in the form 
aul + ßu2 + + 5114 
Find a possible set of values for a, ß, 7, 5. Enter the values of a, B, 7, 5 asa sequence in the box below

 

Machine generated alternative text:
(20* 269: 
(171: -62) 
( -221:-260) 
( 111: -281) 
(4591: 18233: 11908: 6131: 584) : 
(ullu21u31u4) 
205 
269 
264 
-157 
20 
LinearSolve(Å, v) 
171 
164 
263 
-232 
-62 
-54 
-229 
-212 
-221 
-260 
-111 
146 
-281 
192 
-281 
-29 
64 
28

 

Machine generated alternative text:
Suppose that S 
-159 - 206 x 
-286 - 293 - 212 
P3@) — 294 + 245 
P4}, where 
2 
145 
— 224 
2 
— 212 
2 
— 58 x 
— 240 
4 
- 191 x 
4 
- 100 x 
4 
—170 x 
110 — 95 x + 195 x +184 x —60m , and 
-3459 - 7891 + 26850 + 41888 x3 + 17619 
4 
To avoid typing errors; you can copy and past the following sequences to your Maple worksheet 
-159, -206, -145, 
-224, -191 
-286, -293, 
-212, -212, -100 
294, 245, 
-58, -24€, 
110, -95, 195, 184, 
-3459, -7891, 2685€, 
-17e 
-60 
41888, 17619 
The polynomial p is a linear combination of S written in the form 
apl + ßP2 + 7'P3 + 
Find a possible set of values for a, ß, 7, 5. Enter the values of a, B, 7, 5 asa sequence in the box below

 

Machine generated alternative text:
with(LtnearÅ lgebra) 
p. 
( -224:-191): 
(-286:-293: : 
(294224*-58: -240:-170) . 
184:-60) 
(-3459:-7891: 41888: 17619) 
LinearSolve( (pl Ip21p31p4), p) 
11 
-70 
_ 90 
43

 

Machine generated alternative text:
Suppose that S — {111 , 112, 113, 114, 115, 116} C R 
-81 
-40 
-90 
-98 
where 
u2 
-54 
-60 
113 
47 
-80 
81 
-60 
114 
411 
54 
-67 
-58 
115 — 
-40 
69 
-71 
7 
and 116 
-67 
-8 
690 
255 
535 
-305 
To avoid typing errors, you can copy and past the following sequences to your Maple worksheet 
-81, 
-54, 
81, 
¯ 71 , 
69€, 
-40, 
-60, 
-58, 
7, 
255, 
-90, 
47, 
411, 
-4€, 
-67, 
535, 
-98 
-80 
54 
69 
-8 
-3€5 
Since the number of vectors in S is greater that the dimension of R the set S 
Find a possible setof values for Al, A 4, not all zero, such that 
Enter the valuesof Al, .\2, as a sequence in the box below 
must be linearly dependent 
Hint: There are infinitely many solutions for Al, h, 
some nice values for the parameters. 
The solution given by Maple will be in terms of parameters. To get one possible set of values, not all zero, choose

 

Machine generated alternative text:
restart : 
with(LtnearÅ lgebra) 
-98) 
69) 
(-71, 
(ullu21u31u41u51u6) 
GaussianElim inatton( S ) 
-81 
o 
o 
LinearSolve(S (O, O, O, O)) 
-54 
o 
o 
-81 
-54 
-40 
-60 
-90 
47 
-98 
-80 
81 
-100 
o 
o 
81 
-60 
411 
54 
-67 
2018 
81 
61463 
1350 
o 
-67 
-58 
-40 
69 
-71 
-67 
-71 
690 
535 
-305 
3407 
81 
396649 
2700 
71170967 
122926 
690 
2315 
27 
91241 
180 
355854835 
122926

 

Machine generated alternative text:
The column space of a matrix A, col(A) is the span of the columns of A. The rank of A is the dimension of col(A). 
Use Maple to find the dimension of the column space of A, where 
-2290 
-5390 
-2766 
-4822 
2776 
9852 
24444 
11688 
21408 
-12348 
-19137 
-49723 
-23291 
-43959 
25206 
11147 
30233 
14205 
26969 
-15226 
19440 
50848 
24192 
44928 
-25484 
To avoid typing errors; you can copy and past the following sequence of entries of A 
-2290; -5390, -2766, -4822, 2776, 9852, 24444, 11688; 21408, -12348; -19137, -49723, -23291 -43959, 25206, 11147, 30233, 14205; 26969, -15226; 19440, 50848, 24192, 44928, -25484 
and edit it appropriately to make a matrix Enter the rank of A in the box below 
5 
This question accepts numbers or formulae 
Plot I Help Switch to Equation Editor Preview

 

Machine generated alternative text:
restart : 
with(LtnearÅ lgebra) 
Dimension(Å ) 
RowDtm ension(Å ) 
ColumnDtm enston(Å ) 
Rank(Å ) 
((-22902 -5390:-2766: 11688221408: 19137:-49723: -23291:-43959: 1420* 26969: 44928, -25484)) 
-2290 
-5390 
-2766 
-4822 
2776 
9852 
24444 
11688 
21408 
- 12348 
-19137 
-49723 
-23291 
-43959 
25206 
11147 
30233 
14205 
26969 
- 15226 
19440 
50848 
24192 
44928 
-25484

 

 

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