Let A be an n × n matrix and consider the linear operator on Rn defined by L (u) = Au,foru in Rn . A subspace W of Rn is called invariant under L if for any w in W , L (w) is also in W . Show that an eigenspace of A is invariant under L .

Q: Let W be the set of all 2 x 2 matrices A = |" 2 such that Az = 0, where z = H. Is W subspace of M22?…

A: Given that W is the set of all 2×2 matrices A such that Az=0 where A=abcd. Here z=11. We have to…

Q: Let 8 = {1, x, ex, xe*} be a basis for a subspace W of the space of continuous functions, and let Dy…

A:

Q: 1. If T is a normal operator on a finite dimensional inner product space V and if W is a subspace of…

A:

Q: Let V be the vector space of all (3×3) matrices and W be the set of all (3×3) matrices A for which…

A:

Q: Consider L = I – A with A : X → X is a compact linear operator in a normed space X. Show that the…

A: The given problem is related with functional analysis. Given that L = I - A , where A : X → X is a…

Q: Let A be a fixed vector in Rn×n and let S be the set of all matrices that commute with A, that is, S…

A: We have to show that, S = {B | AB = BA}  is a subspace of Rn×n

Q: Let M2x2 be the vector space of real 2 x 2 matrices with entrywise addition and scalar…

A: If 0 is an eigen value of matrix A then it is singular.  Since det(A) is equal to product of eigen…

Q: b] Let W be the set of all 2 × 2 matrices A = such that Az = 0, where z = ||. Is W subspace of M22?…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: Let T be a linear operator on a finite-dimensional vector space V. Prove that the following…

A: An invariant subspace W of T has the property, such that vectors v belongs to W.

Q: Let T be a linear operator on an n-dimensional vector space V such that T has n distinct…

A: Given T:V→V be a linear operator, where V is an n-dimensional vector space such that T has n…

Q: If I is a normal operator on a finite dimensional inner product space V and if W is a subspace of V…

A:

Q: Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on…

A:

Q: Prove that the set of all n x n matrices A such that AT = -A is a subspace of Mn,n.

A: We have to show that set of all skew symmetric matrix form subspace of Mnn

Q: 1. Suppose that A is an nxn non-zero, real matrix and 2 is a fixed real number. Let E ={x=R* : AT =…

A:

Q: (a) Prove that if A and B are similar n x 11 matrices, then tr(A) = tr(B).  (b ) How would you…

A: We are given two matrices A and B and they are similar. We are asked to proved tr(A)=tr(B). Next we…

Q: : Consider the following statements. (1) If A is invertible and -1 is an eigenvalue of A then -1 is…

A:

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: We write y = p+o, where  is the projection of y, on W and o = y −p is orthogonal to W Since  are…

Q: Let w E R3 be a fixed vector with ||w|| = 1 and let f : R³ → R³ be the linear transformation given…

A: Given w∈ℝ3 be a fixed vector with w=1. Let f:ℝ3→ℝ3 be the linear transformation given by:…

Q: 2. Consider the vector space M. [:] a b (a) Show that the set of all matrices of the form where a –…

A:

Q: Let V be the vector space of all 2 × 2 matrices over the real field R. Show that Wis not a subspace…

A: Given, V be the vector space of all 2×2 matrices over the real field R. To show that W is not a…

Q: Let T be a linear operator on a vector space with a T-invariant subspace W.Prove that if x is an…

A:

Q: Suppose the set ​{u​, v​} forms a basis for a subspace W in Rn.  Let x​=u​+v and y​=u​-v.  Show that…

A:

Q: Let T be a linear operator on a vector space with a T-invariant subspace W. Prove that if v is an…

A:

Q: If A is an n x n matrix such that A = - A' and all of the eigenvalues of A are 0 or 1, then A is the…

A:

Q: If W is a 2 x 2 matrices A with trace equal to zero is a subspace of M2,2. Prove that span X = W.…

A:

Q: Let W be the set of all 3 x 3 matrices whose trace is zero. Show that S is a subspace of M33.

A: Let W be the set of all 3×3 matrices whose trace is zero. Show that W is a subspace of M33.

Q: Let V be a vector space and T : V →V a linear transformation. Show that if T" = 0 for somen > 0,…

A: Given: T:V→V is a linear transformation and Tn=0 for some n>0. To show:  0 is an eigenvalue of T…

Q: Let V be a finite-dimensional vector space and T be the projection on W along W’, where W and W’are…

A:

Q: 4. Let be B = {1,eª, xe²} a basis of a subspace E of the space of continuous functions, and let D.…

A:

Q: Let T be a linear operator on a vector space V, and let W1,W2, . . . ,Wk be T-invariant subspaces of…

A:

Q: Let A be an mxn matrix. Prove that every vector x in R" can be written in the form x = p+u, where p…

A: Let A be an m×n matrix.Prove: for evert vector x∈ℝn can be written in the form x=p+u, where p is in…

Q: (4) Let V be a finite-dimensional vector space over F, let T E L(V), and let W be a subspace of V…

A:

Q: Let B be a fixed mxn matrix and let S be the set of nxp matrices where S={A|BA=0}. Determine if S is…

A:

Q: Let P be the projection matrix corresponding to a subspace S of Rm. Show that P2 = P

A: Given :Let p be the projection matrix corresponding to a subspace             S of RM

Q: Let T and T, be linear operators on V and let W be a subspace of V. Show that if W is invariant…

A: Let T1 and T2 be linear operators on V and let W be a subspace of V. If W is invariant under both…

Q: Given two n xn matrices A and B, prove that the set S = {x € R" : ABx = 0} is a subspace of R".

A:

Q: Let V be a finite-dimensional vector space, and let T be a projection on a subspace W of V. Find an…

A: It is given that Let V be a finite-dimensional vector space, and let T be a projection on a subspace…

Q: Let A be a non-singular matrix, det(A) ± 0. a) Show that µ(A) does not change if the operator A is…

A:

Q: Suppose W is a set of 2 x 2 matrices A with trace A = 0 , and is a subspace of M2,2 Consider the…

A:

Q: а) The subspace H =Span- in R³ has dimension b) If A is a 2 x 2 matrix, and the product A =3: then…

A: a) We will evaluate number of linearly independent vectors in H Let x = 100 y = 400 Clearly,  y =…

Q: Let Mnn be a vector space of n x n matrices. Which of the following is not a subspace of Mnn?

A: The vector space of nxn matrix is Mnn.

Q: (a) Let T be a linear operator on a vector space V, and let y be a cycle of generalized eigenvectors…

A: As you have asked more questions, we are solving first question only as per the guidelines. If you…

Q: 4. Let V be a vector space and let L : V → V be a linear transformation. Let A be a scalar. (a)…

A:

Q: An m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero,…

A: Proof:

Q: Let {u1, u2,..., Un} be an orthonormal basis for a subspace S of an inner product space V, and let…

A: The solution is given below

Q: Let B = {1, x, ex, xex} be a basis for a subspace W of the space of continuous functions, and let Dx…

A: Given: B=1, x, ex, x ex is a basis g=for a subspace W of the space of continuous functions. DX is…

Q: Let A be an n×n matrix and let λ be a nonzero eigenvalue of A. Show that if x is an eigenvector…

A: Given: Let A denote n×n matrix and λ be a non zero eigen value of A. Thus, A is an n×n matrix with…

Q: Let A be an mxn matrix. Prove that W = {x ER" | Ax = 0} is a subspace of R", where W is the solution…

A:

Q: Let A be an n×n matrix whose characteristic polynomial splits, γ be a cycle of generalized…

A: (a) This can be done as follows:

Q: Let A be an n × n matrix. Show that a vector x in either Rn or Cn is an eigenvector belonging to A…

A: According to the given information, let A be an n×n matrix. It is required to show that a vector x: