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5 1 0 Problem 4: LetA = 0 41 . Consider the linear operator LA : R3 → R3 a) Find the characterist...
Let T: C4 C4 be the linear operator such that 18= [ 0 1 0 1]...
Let T: C4 C4 be the linear operator such that 18= [ 0 1 0 1] 1 1 1 0 1-100 i Oi i 1 where 8 is the standard basis for C4. Let W be the T-cyclic subspace of C4 generated by w = (1,0,0,0). 3.1 Find the T-cyclic basis for W generated by w. 3.2 Find a basis for Wt. 3.3 Show that wt is T*- invariant.
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such...
Problem 4. Give an example of a linear operator T on a
finite-dimensional vector space such that T is not nilpotent, but
zero is the only eigenvalue of T. Characterize all such
operators.
Problem 5. Let A be an n × n matrix whose characteristic
polynomial splits, γ be a
cycle of generalized eigenvectors corresponding to an
eigenvalue λ, and W be the subspace spanned
by γ. Define γ′ to be the ordered set obtained from γ by
reversing the...
1. LetA-Lind the follwing a) lA 2. Use expansion by cofactors to find the determinant of...
1. LetA-Lind the follwing a) lA 2. Use expansion by cofactors to find the determinant of the matrix. A 4 or column that you are expanding.) 5 0(In your solution, state the row -3 6-4 3. Let u (1,-2,4,-5), (8,-10, -2,3) and w (1,0,8,0). Find the following a.) 6u 4. If possible, write vas a linear combination of ul, u2 and ug. ii! = (4,3,-2) , iz (8,6,1), u,-(-4,5,12), U = (4,-13,-17) 5. Let Wbe the set of all 3...
4. Let T be a linear operator on the finite-dimensional space V with eharacteristie polynomial and minimal polynomial Let W be the null space of (T c) Elementary Canonical Forms Chap. 6 226 (a)...
4. Let T be a linear operator on the finite-dimensional space V with eharacteristie polynomial and minimal polynomial Let W be the null space of (T c) Elementary Canonical Forms Chap. 6 226 (a) Prove that W, is the set of all vector8 α in V such that (T-cd)"a-0 for some positive integer 'n (which may depend upon α). (b) Prove that the dimension of W, is di. (Hint: If T, is the operator induced on Wi by T, then...
(1 point) Consider the multiplication operator LA: RR4 where 13-ї 8 -5 9 16 A- 0 0 02J Find a matrix B whose row space is smallest LA-invariant subspace that contains the vector (1,0,-1,0). (1 p...
(1 point) Consider the multiplication operator LA: RR4 where 13-ї 8 -5 9 16 A- 0 0 02J Find a matrix B whose row space is smallest LA-invariant subspace that contains the vector (1,0,-1,0).
(1 point) Consider the multiplication operator LA: RR4 where 13-ї 8 -5 9 16 A- 0 0 02J Find a matrix B whose row space is smallest LA-invariant subspace that contains the vector (1,0,-1,0).
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) F...
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) Find the monic generators of the ideals S(u;W), S(q;W), s(G;W), 1
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W...
(22). (10 Marks). Consider vectors v = 0 - in R3 (a). (5 marks) Find a...
(22). (10 Marks). Consider vectors v = 0 - in R3 (a). (5 marks) Find a basis of the subspace of R3 consisting of all vectors perpendicular to v. (b). (5 marks) Find an orthonormal basis of this subspace.
Linear Algebra Problem! 1. Let U be the subspace of R3 given by 11 + 12...
Linear Algebra Problem!
1. Let U be the subspace of R3 given by 11 + 12 - 213 = 0. for U. Justify that is an ordered basis. What is the a) Find an ordered basis dimension of U? b) Let ū= (1,1,1). Show that ✓ EU and find the B-coordinate vector (Ū3 = C:(Ū), where Ce: U + R2 is the B-coordinate transformation.
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-x...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
Problem 4 Let V be the vector space of functions of...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0},...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
Let T: C4 C4 be the linear operator such that 18= [ 0 1 0 1] 1 1 1 0 1-100 i Oi i 1 where 8 is the standard basis for C4. Let W be the T-cyclic subspace of C4 generated by w = (1,0,0,0). 3.1 Find the T-cyclic basis for W generated by w. 3.2 Find a basis for Wt. 3.3 Show that wt is T*- invariant.
Problem 4. Give an example of a linear operator T on a
finite-dimensional vector space such that T is not nilpotent, but
zero is the only eigenvalue of T. Characterize all such
operators.
Problem 5. Let A be an n × n matrix whose characteristic
polynomial splits, γ be a
cycle of generalized eigenvectors corresponding to an
eigenvalue λ, and W be the subspace spanned
by γ. Define γ′ to be the ordered set obtained from γ by
reversing the...
1. LetA-Lind the follwing a) lA 2. Use expansion by cofactors to find the determinant of the matrix. A 4 or column that you are expanding.) 5 0(In your solution, state the row -3 6-4 3. Let u (1,-2,4,-5), (8,-10, -2,3) and w (1,0,8,0). Find the following a.) 6u 4. If possible, write vas a linear combination of ul, u2 and ug. ii! = (4,3,-2) , iz (8,6,1), u,-(-4,5,12), U = (4,-13,-17) 5. Let Wbe the set of all 3...
4. Let T be a linear operator on the finite-dimensional space V with eharacteristie polynomial and minimal polynomial Let W be the null space of (T c) Elementary Canonical Forms Chap. 6 226 (a) Prove that W, is the set of all vector8 α in V such that (T-cd)"a-0 for some positive integer 'n (which may depend upon α). (b) Prove that the dimension of W, is di. (Hint: If T, is the operator induced on Wi by T, then...
(1 point) Consider the multiplication operator LA: RR4 where 13-ї 8 -5 9 16 A- 0 0 02J Find a matrix B whose row space is smallest LA-invariant subspace that contains the vector (1,0,-1,0).
(1 point) Consider the multiplication operator LA: RR4 where 13-ї 8 -5 9 16 A- 0 0 02J Find a matrix B whose row space is smallest LA-invariant subspace that contains the vector (1,0,-1,0).
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) Find the monic generators of the ideals S(u;W), S(q;W), s(G;W), 1
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W...
(22). (10 Marks). Consider vectors v = 0 - in R3 (a). (5 marks) Find a basis of the subspace of R3 consisting of all vectors perpendicular to v. (b). (5 marks) Find an orthonormal basis of this subspace.
Linear Algebra Problem!
1. Let U be the subspace of R3 given by 11 + 12 - 213 = 0. for U. Justify that is an ordered basis. What is the a) Find an ordered basis dimension of U? b) Let ū= (1,1,1). Show that ✓ EU and find the B-coordinate vector (Ū3 = C:(Ū), where Ce: U + R2 is the B-coordinate transformation.
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
Problem 4 Let V be the vector space of functions of...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
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