Maple1231W6T5 - Further Linear Algebra II

Sunday, 2 September 2018

7:03 PM

Untitled picture.png Machine generated alternative text:
You should work through the "Lesson 10 - Further Linear Algebra" self-paced learning module on UNSW Moodle_ 
115 — 
You may need the command Rank in the package LinearA1gebra_ 
Suppose that S = {111 , 112, 113, 114, 115, 116} C R 
111 — 
83 
-57 
-64 
-47 
where 
u2 
-42 
90 
113 
-65 
-56 
1 
-123 
114 
194 
159 
-26 
40 
26 
-46 
89 
-62 
-19 
-93 
and 116 
-319 
266 
109 
187 
To avoid typing errors, you can copy and past the following sequences to your Maple worksheet 
83, 
-57, 
-42, 90, 
1, -123, 
-26, 40, 
89, -62, 
-64, 
-65, 
194, 
26, 
-19, 
-47 
-56 
159 
-46 
-93 
-319, 266, 1€9, 187 
Let A be the 4 x 6 matrix whose columns are vectors of S in order. 
The rank of A is the dimension ot span(S) 
If the rank of A is equal to dim(R4), then S 
O 
If the rank of A is less than dim(R then S 
O 
4 
a spanning set for R 
4 
a spanning set for R 
Use Maple to find the rank of A and enter your answer in the box below 
Hence; S 
4 
is 
O 
4 
a spanning set for R 

Untitled picture.png Machine generated alternative text:
.. , vn} C Rm Let A be the m x n matrix whose columns are vectors of S in order. 
Suppose that S = {VI' • 
The rank of A is the dimension ot span(S) ' 
If the rank of A is equal to the number of vectors in S then S 
If the rank of A is less than the number of vectors in S, then S 
is 
is not 
O 
O 
linearly independent 
linearly independent 
Untitled picture.png Machine generated alternative text:
u6 : 
-47) 
-56) 
159) 
-93) 
187) 
(ullu21u31u41u51u6) 
83 
-64 
-47 
-42 
90 
-65 
-56 
-123 
194 
159 
4 
-26 
40 
26 
_ 46 
89 
-62 
-19 
-93 
-319 
266 
109 
187 
with(LtnearÅ lgebra) 
Rank(Å )
Untitled picture.png Machine generated alternative text:
.. , vn} C Rm Let A be the m x n matrix whose columns are vectors of S in order. 
Suppose that S = {VI' • 
The rank of A is the dimension ot span(S) ' 
If the rank of A is equal to the number of vectors in S then S 
If the rank of A is less than the number of vectors in S, then S 
is 
is not 
O 
O 
linearly independent 
linearly independent 

Untitled picture.png Machine generated alternative text:
(a) Suppose that A is an m x n real matrix. The function TA : IRn —+ Rm 
is a linear transformation. 
defined by 
T A (x) = Ax, for all x e IRn 
6 
-92 
17 
59 
-31 
66 
-94 
87 
-23 
82 
65 
86 
-36 
-81 
-64 
63 
-36 
-86 
and x 
-77 
99 
-6 
18 
-10 
-19 
12 
Then TA (x) 
To avoid typing errors, you can copy and past the following sequence of entries of A 
6, -92, 17, 59, -31, 66, -94, 87; -23, 82, 65, 86, -36, -81, -64, 63; -36; -86; -77; 99 
Recall: the Maple notation for a vector 
b 
is < a,b,c 
Alternatively you can copy your answer from your Maple worksheet and paste it to the answer box 
(b) Suppose now that the linear map T : IR3 IR2 is defined by, for all x 
12 — 7 
3 
Enter the matrix M , in Maple syntax, in the box below such that T(x) = Mx for all x e R 
a 
Recall: the Maple notation for a matrix 
d 
b 
c 

Untitled picture.png Machine generated alternative text:
Let A be an m x n matrix. The kernel of A is the setof vectors 
ker(A) = {x 
which is a vector space. 
The dimension of ker(A) is called the nullity of A, denoted by nullity(A) 
(a) Find the nullity of the matrix 
168 
-22 
90 
78 
and enter in the box below 
nullity(A) — 
To avoid typing errors, you may copy and paste the following sequence of entries 
168; -22, 90, 78, -16, 57, 201, 125, 26; 48, 106, -112; -126; 20, 50, 198; -120, -135, -223, 573 
-16 
57 
201 
125 
26 
48 
106 
-112 
-126 
20 
50 
198 
-120 
-135 
-223 
573 
Untitled picture.png Machine generated alternative text:
with(LtnearÅ lgebra) 
-92: -81, -86: 
-77:99)) 
Multiply(Å, (-6, 18, 
6 
-92 
17 
59 
-10,-19, 12)) 
-31 
66 
-94 
87 
-23 
82 
65 
86 
-112 
1427 
-2152 
343 
-36 
-81 
-64 
63 
-36 
-86 
-77 
99 
(2)
Untitled picture.png Machine generated alternative text:
Let A be an m x n matrix. The kernel of A is the setof vectors 
ker(A) = {x 
which is a vector space. 
The dimension of ker(A) is called the nullity of A, denoted by nullity(A) 
(a) Find the nullity of the matrix 
168 
-22 
90 
78 
and enter in the box below 
nullity(A) — 
To avoid typing errors, you may copy and paste the following sequence of entries 
168; -22, 90, 78, -16, 57, 201, 125, 26; 48, 106, -112; -126; 20, 50, 198; -120, -135, -223, 573 
-16 
57 
201 
125 
26 
48 
106 
-112 
-126 
20 
50 
198 
-120 
-135 
-223 
573 

Untitled picture.png Machine generated alternative text:
(b) For the matrix A in (a); select all the statements below which are true. 
0 
is in ker(Ab 
is in ker(Ab 
Cl ker(A) is subspace of IR4 
is in ker (Ab 
ker(A) is subspace of 
-50 
34 
78 
43 
is in ker(A) 
is in ker(A). 
You may copy and paste the entries 
-50, 34, 78. 43 
50 
-68 
is in ker(A). 
You may copy and paste the entries 
50, -68, 74 
Untitled picture.png Machine generated alternative text:
-135, -223,573)) 
Dimension(Å ) 
RowDtm ension(Å ) 
ColumnDtm enston(Å ) 
Rank(Å ) 
RowSpace(Å ) 
100 
ColumnSpace(Å ) 
with(LtnearÅ lgebra) 
((168, -22,902 120: 
-4 
168 
-22 
90 
78 
o 
2 
-16 
57 
201 
125 
o 
26 
48 
106 
-112 
- 126 
20 
198 
4 
1011 
802089 
120074 
2 
4 
2 
- 120 
-135 
-223 
573 
286011 
120074 
202507 
120074 
NullSpace(Å ) 
nops (NullSpace(Å ) ) 
Rank(Å ) ) ) 
ColumnDtm eras ton(Å ) ; 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11)
Untitled picture.png Machine generated alternative text:
(b) For the matrix A in (a); select all the statements below which are true. 
0 
is in ker(Ab 
is in ker(Ab 
Cl ker(A) is subspace of IR4 
is in ker (Ab 
ker(A) is subspace of 
-50 
34 
78 
43 
is in ker(A) 
is in ker(A). 
You may copy and paste the entries 
-50, 34, 78. 43 
50 
-68 
is in ker(A). 
You may copy and paste the entries 
50, -68, 74 

Untitled picture.png Machine generated alternative text:
Let A be an m x n matrix. The image of A is the setof vectors 
im(A) 
which is a vector space. 
The dimension of im(A) is called the rank of A, denoted by rank(A) 
(a) Find the rank of the matrix 
and enter in the box below 
Ax for some x e IR n} 
108 
4 
198 
170 
-48 
122 
75 
-23 
178 
-130 
118 
190 
3 
-280 
250 
-142 
-298 
-760 
642 
-397 
-783 
rank(A) 
Note: The Maple command for find the rank of A is Rank(A), upper case R. 
To avoid typing errors, you may copy and paste the following sequence of entries 
108; 4; 198, 170; -48; 122, 75, -23, 178; -130; 118, 190, -280, 250, -142, -298, -760; 642, -397, -783 

Untitled picture.png Machine generated alternative text:
(b) For the matrix A in (a); select all the statements below which are true. 
-51 
60 
is in im(A). 
-12 
You may copy and paste the entries 
-51, 60, -12, -54 
is in im(Ab 
-76 
25 
is in im(A). 
17 
-40 
You may copy and paste the entries 
0 
is in im(A) 
is in im(Ab 
im(A) is subspace of IR4 
Cl im(A) is subspace of 
is in im(A) 
Untitled picture.png Machine generated alternative text:
-135, -223,573)) 
Dimension(Å ) 
RowDtm ension(Å ) 
ColumnDtm enston(Å ) 
Rank(Å ) 
RowSpace(Å ) 
100 
ColumnSpace(Å ) 
with(LtnearÅ lgebra) 
((168, -22,902 120: 
-4 
168 
-22 
90 
78 
o 
2 
-16 
57 
201 
125 
o 
26 
48 
106 
-112 
- 126 
20 
198 
4 
1011 
802089 
120074 
2 
4 
2 
- 120 
-135 
-223 
573 
286011 
120074 
202507 
120074 
NullSpace(Å ) 
nops (NullSpace(Å ) ) 
Rank(Å ) ) ) 
ColumnDtm eras ton(Å ) ; 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 
Untitled picture.png Machine generated alternative text:
Dimension(Å ) 
RowDtm ension(Å ) 
ColumnDtm enston(Å ) 
Rank(Å ) 
RowSpace(Å ) 
100 
ColumnSpace(Å ) 
with(LtnearÅ lgebra) 
122: 75, -130, 118, 250:-142: 
-783)) 
-4 
108 
4 
198 
170 
o 
2 
-48 
122 
-23 
o 
178 
-130 
118 
190 
-280 
250 
- 142 
-298 
4 
1011 
16880 
15851 
2 
4 
2 
- 760 
642 
-397 
- 783 
001 
17929 
15851 
7294 
NullSpace(Å ) 
nops (NullSpace(Å ) ) 
Rank(Å ) ) ) 
ColumnDtm enston(Å ) ; 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11)
Untitled picture.png Machine generated alternative text:
(b) For the matrix A in (a); select all the statements below which are true. 
-51 
60 
is in im(A). 
-12 
You may copy and paste the entries 
-51, 60, -12, -54 
is in im(Ab 
-76 
25 
is in im(A). 
17 
-40 
You may copy and paste the entries 
0 
is in im(A) 
is in im(Ab 
im(A) is subspace of IR4 
Cl im(A) is subspace of 
is in im(A) 
Untitled picture.png Machine generated alternative text:
Dimension(Å ) 
RowDtm ension(Å ) 
ColumnDtm enston(Å ) 
Rank(Å ) 
RowSpace(Å ) 
100 
ColumnSpace(Å ) 
with(LtnearÅ lgebra) 
122: 75, -130, 118, 250:-142: 
-783)) 
-4 
108 
4 
198 
170 
o 
2 
-48 
122 
-23 
o 
178 
-130 
118 
190 
-280 
250 
- 142 
-298 
4 
1011 
16880 
15851 
2 
4 
2 
- 760 
642 
-397 
- 783 
001 
17929 
15851 
7294 
NullSpace(Å ) 
nops (NullSpace(Å ) ) 
Rank(Å ) ) ) 
ColumnDtm enston(Å ) ; 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11)

 

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