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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