Prove the trace of an n x n matrix is an element of the dual space...

Prove the trace of an n x n matrix is an element of the dual space of all n x n matrices.

Expert Solution

First we show that trace is a linear function from set of all matrices of order n and field . then it must be an element of dual space.

Colby Messinger answered 2 hours ago

Problem 1 1.1 If A is an n x n matrix, prove that if A has...

Problem 1 1.1 If A is an n x n matrix, prove that if A has n linearly independent eigenvalues, then AT is diagonalizable. 1.2 Diagonalize the matrix below with eigenvalues equal to -1 and 5. 0 1 1 2 1 2 3 3 2 1.3 Assume that A is 4 x 4 and has three different eigenvalues, if one of the eigenspaces is dimension 1 while the other is dimension 2, can A be undiagonalizable? Explain. Answer for all...

1. For an m x n matrix A, the Column Space of A is a subspace...

1. For an m x n matrix A, the Column Space of A is a subspace of what vector space? 2. For an m x n matrix A, the Null Space of A is a subspace of what vector space?

Prove that the dual space of l^1 is l^infinity

Let A be an m x n matrix. Prove that Ax = b has at least...

Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

Prove: An (n × n) matrix A is not invertible ⇐⇒ one of the eigenvalues of...

Prove: An (n × n) matrix A is not invertible ⇐⇒ one of the eigenvalues of A is λ = 0

trace element

Is a trace element an essential element? Explain. Explain.

A m*n matrix A. P is the dimension of null space of A. What are the...

A m*n matrix A. P is the dimension of null space of A. What are the number of solutions to Ax=b in these cases. Prove your answer. a. m=6, n=8, p=2 b. m=6, n=10, p=5 c. m=8, n=6, p=0

1. Prove that the Cantor set contains no intervals. 2. Prove: If x is an element...

1. Prove that the Cantor set contains no intervals. 2. Prove: If x is an element of the Cantor set, then there is a sequence Xn of elements from the Cantor set converging to x.

Let x = inf E. Prove that x is an element of the boundary of E....

Let x = inf E. Prove that x is an element of the boundary of E. Here R is regarded as a metric space with d(p,q) = |q−p| for p,q ∈R

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Question