A square matrix A is nilpotent if A" : = 0 for some positive integer n. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty? choose 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer [1 2] [56] 3.6 . (Hint: to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A 3 4 and B such that (A + B)" ‡ 0 for all positive integers n.) 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer [34] 2, 3] (Hint: to show that is not closed under scalar multiplication, it is sufficient to find a real number r 5 and a nilpotent matrix A such that (rA)” ‡ 0 for all positive integers n.) 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose

Q: Assume that T is a linear transformation. T: Projects vectors in its domain onto the line y = 2/3 x.…

A:

Q: Determine whether the set of all 2 x 2 nonsingular matrices, together with matrix addition and…

A: See the solution below.

Q: If A and B are symmetric matrices, will A*B*A be symmetric? Please provide proof.

A: Concepts to be used :A matrix Mis said to be symmetric if,MT=M, where MTis the transpose of the…

Q: In linear algebra, if a vector is in the range of a matrix A, then it is also in the null space of…

A: The given statement is False.

Q: Г1 2 3 4 5 A = 5 0 2 4 6 8 5 5 5 5

A:

Q: Prove using elements that for any square matrices A,B of the same size and constant K.…

A: For any square matrices A,B of the same size and constant K.  tr(kA+B)=ktr(A)+tr(B)

Q: Which of the following is not a vector space?    A. The set of all 2x2 with integer entries and…

A:

Q: Consider the set of all 3x3 matrices. Prove that with standard addition and standard multiplication…

A: The given question is solved below.

Q: Let X = TONT. 1 and 3 0 A = 0 3 1 0 -2 -4 -4 0 3 00 -2 0 0 -4 0 1 -1 01 1 0 0

A:

Q: TRUE/FALSE Assume that all the given matrices are n × n. If A = O, but Ak = O (where O denotes the…

A: Given:   To Check: A is singular or not Concept: A is singular only when det(A) = 0

Q: Let A = 3 ů 5. positive definite show that the matrix A is. پول و مع دول ۷۵

A: A matrix is said to be positive definite if the given matrix is symmetric and all its eigen values…

Q: Let ¹ = [ 8 ], x= [ ¦ ! ] x = [ 9 ] -[69]-[88]-[8][9] Show that these four matrices span M2(C), the…

A:

Q: Let A and B be two skew-symmetric matrices of the same size. Show that AB+BA is symmetric.

A: Given that: Let A and B be two skew-symmetric matrices of the same size. To show: AB+BA is…

Q: EXER 7: MATRICES Prove that if N is invertible then MN¹ = N¹M if and only if MN=NM.

A:

Q: Exhibit a basis for M2x2, the vector space of all 2 x 2 matrices of real numbers.

A:

Q: Let A be a 7 x 4 matrix and let v be a vector with 4 entries. Which of the following statements is…

A: as we know that when a matrix is multiplied by a vector then the product is a vector.

Q: Basis Find a basis for the space of 2 x 2 lower triangular matrices.

A: In this question, we have to find the basis for the space of 2×2 lower triangular matrices. We know…

Q: Which of the following is not a vector space? (; :) A. The set of all 2x2 matrices of the form with…

A: Solution:-

Q: Does the set of all 2x2 matrices form a vector space, please explain.

A:

Q: Let u be a vector in R" and A be an m × n matrix. If u is orthogonal to each row of A then u is in…

A: Since u is orthogonal to each row of A, it means u is in null space of A.

Q: The set of all 2x2 symmetric matrices is a vector space Select one: True False

A:

Q: Consider the vector space PA(R) = {ax + bæ³ + ca? + dx + e : a, b, c, d, e E R} of polynomials of…

A: Note:- for this problem you don't mention about f1, f2, f3 (should i have to take  f1, f2, f3…

Q: Determine the dimensions of Nul A and Col A for the matrices. A =

A:

Q: Suppose there exists a 2x2 diagonal matrix with a standard operation. Prove that this set is a…

A:

Q: 47. Prove that if A and B are square matrices and AB is invertible, then both A and B are…

A:

Q: Let V be the set of all 2 × 3 matrices. Verify that V is a vector space.

A:

Q: 2- Let v toe with real entries · Then Check Ų formes ailo Vector space or not the set of all 2x2…

A:

Q: List the Invertible Matrix Theorem statements that can be either all true or all false

A:

Q: Let A be a 5 x 5 matrix such that each row of A contains every mumber from the following list…

A: From given matrix

Q: The set of all 2x2 symmetric matrices is a vector space Select one: O True O False

A: Note the result : V :Set of all 2×2 matrices is a vector space over ℝ . A : Set of all symmetric…

Q: Let U be the space of 3 x 3 symmetric matrices and V be the space of 3 x 3 skew symmetric matrices.…

A: We have to solve about dimension of symmetric and skew symmetric matix