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(1 point) The linear transformation T: R4 R4 below is diagonalizable. T(x,y,z,w) = (x – -...
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no...
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no credit. (1) A square matrix with the characteristic polynomial X - 413 +212 - +3 is invertible. [23] (II) Matrix in Zs has two distinct eigenvalues. (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices Dand D2 are...
Please provide answer in neat handwriting. Thank you Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operato...
Please provide answer in neat handwriting. Thank you
Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
Ler L: R4 → R3 be the linear transformation defined by (4p) L(z,y,z, t) = (x...
Ler L: R4 → R3 be the linear transformation defined by (4p) L(z,y,z, t) = (x – y +t, 2x – 2, Y + 2z – t) a) Find the images of the standard basis of RA L(1,0,0,0) = L(0,1,0,0) = L(0,0,1,0) = L(0,0,0,1) = b) Find a basis and the dimension of the image of L c) Find a basis and the dimension of the kernel of L (8p) (8p)
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal. 12: Find a basis B for R', such th...
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a +...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(....
s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(. :) : 6 1) (1 1) (1 :) :()} is linearly independent. (e) (1 point) For a linear transformation A:R" + Rd the dimension of the nullspace is larger than d. (f) (1 points) Let AC M4x4 be a diagonal matrix. A is similar to a matrix A which has eigenvalues 1,2,3 with algebraic multiplicities 1,2, 1 and geometric multiplicities 1,1, 1 respectively. 8....
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no credit. (1) A square matrix with the characteristic polynomial X - 413 +212 - +3 is invertible. [23] (II) Matrix in Zs has two distinct eigenvalues. (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices Dand D2 are...
Please provide answer in neat handwriting. Thank you
Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
Ler L: R4 → R3 be the linear transformation defined by (4p) L(z,y,z, t) = (x – y +t, 2x – 2, Y + 2z – t) a) Find the images of the standard basis of RA L(1,0,0,0) = L(0,1,0,0) = L(0,0,1,0) = L(0,0,0,1) = b) Find a basis and the dimension of the image of L c) Find a basis and the dimension of the kernel of L (8p) (8p)
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(. :) : 6 1) (1 1) (1 :) :()} is linearly independent. (e) (1 point) For a linear transformation A:R" + Rd the dimension of the nullspace is larger than d. (f) (1 points) Let AC M4x4 be a diagonal matrix. A is similar to a matrix A which has eigenvalues 1,2,3 with algebraic multiplicities 1,2, 1 and geometric multiplicities 1,1, 1 respectively. 8....
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