1. Let ??(F), ??(F) and ??(F) denote the spaces of strictly upper triangular, diagonal, and strictly...

1. Let ??(F), ??(F) and ??(F) denote the spaces of strictly upper triangular, diagonal,

and strictly lower triangular ?×? matrices over F, respectively. Show that ??(F) = ??(F)⊕??(F)⊕??(F). Compute the dimensions of these spaces and show they sum

2 up to ?.

2. Is the set {?3 +2?2,−?2 +3?+1,?3 −?2 +2?−1} in P3(R) linearly independent? Is it a basis?

Solutions

Expert Solution

Colby Messinger answered 2 years ago

Next > < Previous

Related Solutions

et A be a 157 x 157 upper-triangular matrix. Suppose that every diagonal entry of A...

et A be a 157 x 157 upper-triangular matrix. Suppose that every diagonal entry of A is 1 and that there is at least one nonzero off-diagonal entry in A. Is A diagonalizable? Explain how you can answer this question mentally, with no non-trivial calculations.

1) Let A be nxn matrix and Ax=b, if we need change A to Upper triangular...

1) Let A be nxn matrix and Ax=b, if we need change A to Upper triangular matrix using Gaussian Elimination, how many additions/subtraction operations are involved? how many multiplication/division operations are involved? 2) Once we got the upper triangular matrix, now we need to apply back-substitution, how many additions/subtraction operations are involved? how many multiplication/division operations are involved?

Question 1. Let V and W be finite dimensional vector spaces over a field F with...

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry along the diagonal is either a 0 or a 1. Hint 1. Suppose A = {~v1, . . . , ~vn}...

Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ...

Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ second on a computing resource, compute the time taken to solve the system Ux = b, assuming it has a unique solution. What would be the time taken if Un×n is lower triangular

Let Vand W be vector spaces over F, and let B( V, W) be the set...

Let Vand W be vector spaces over F, and let B( V, W) be the set of all bilinear forms f: V x W ~ F. Show that B( V, W) is a subspace of the vector space of functions 31'( V x W). Prove that the dual space B( V, W)* satisfies the definition of tensor product, with respect to the bilinear mapping b: V x W -> B( V, W)* defined by b(v, w)(f) =f(v, w), f E...

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y...

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y be a continuous onto map with the property that f-1[{y}] is connected for every y∈Y. Show that ifY is connected then so isX.

(A universal random number generator.)Let X have a continuous, strictly increasing cdf F. Let Y =...

(A universal random number generator.)Let X have a continuous, strictly increasing cdf F. Let Y = F(X). Find the density of Y. This is called the probability integral transform. Now let U ∼ Uniform(0,1) and let X = F−1(U). Show that X ∼ F. Now write a program that takes Uniform (0,1) random variables and generates random variables from an Exponential (β) distribution

Let Z denote the set of all integers. Give an explicit bijection f : Z →...

Let Z denote the set of all integers. Give an explicit bijection f : Z → N

Prove the following. Let T denote the integers divisible by three. Find a bijection f :...

Prove the following. Let T denote the integers divisible by three. Find a bijection f : Z→T (Z denotes all integers).

Let f : V mapped to W be a continuous function between two topological spaces V...

Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V...

Subjects