Question

Let E be an n×n matrix, and letU= {xE:x∈Rn} (where x∈Rn is written as arow vector)....

Let E be an n×n matrix, and letU= {xE:x∈Rn} (where x∈Rn is written as arow vector). Show that the following are equivalent.
(a) E^2 = E = E^T (T means transpose).
(b) (u − uE) · (vE) = 0 for all u, v ∈ Rn.

(c) projU(v) = vE for all v ∈ Rn.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Problem 3.2 Let H ∈ Rn×n be symmetric and idempotent, hence a projection matrix. Let x...
Problem 3.2 Let H ∈ Rn×n be symmetric and idempotent, hence a projection matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number. Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and find the joint distribution of (u,v). 1 (c) Someone claims that u and v are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ. (e) Assume that 1 ∈ Im(H) and find...
Let A be an m×n matrix, x a vector in Rn, and b a vector in...
Let A be an m×n matrix, x a vector in Rn, and b a vector in Rm. Show that if x1 in Rn is a solution to Ax=b and x2 is a solution to Ax=⃗0, then x1 +x2 is a solution to Ax=b.
Let x be a vector in Rn written as a column, and define U = {Ax:A∈Mmn}....
Let x be a vector in Rn written as a column, and define U = {Ax:A∈Mmn}. Show that U is a subspace of Rm.
Let X be a not-random nxk Matrix. Let Y=Xbeta +u, with E(u)=0 a Vector beta is...
Let X be a not-random nxk Matrix. Let Y=Xbeta +u, with E(u)=0 a Vector beta is a true value if E(y)=Xbeta Show that two solutions beta1_hat and beta2_hat of the normal equations fulfill the following equation Xbeta1_hat=Xbeta2_hat if rank(X)=k, show that the normal equations have a unique solution Normal equations: XtXbeta=XtY
Let U, V be a pair of subspaces of Rn and U +V the summationspace. Suppose...
Let U, V be a pair of subspaces of Rn and U +V the summationspace. Suppose that U ∩ V = {0}. Prove that from every vector U + V can be written as the sum of a vector from U and a vector from V.
n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix...
n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix theorem below and show that all 3 statements are true or false. Make sure to clearly explain and justify your work. A= -1 , 7, 9 7 , 7, 10 -3, -6, -4 The equation A has only the trivial solution. 5. The columns of A form a linearly independent set. 6. The linear transformation x → Ax is one-to-one. 7. The equation Ax...
Let E be the mxm matrix that extracts the "even part" of an m-vector; Ex=(x+Fx)/2 where...
Let E be the mxm matrix that extracts the "even part" of an m-vector; Ex=(x+Fx)/2 where F is the mxm matrix that flips {x1, ...,xm}* to {xm, ..., x1}*. What is (Ex)^2. I'm lost on how to calculate this.
Let A be an n × n matrix, v a column vector, and suppose {v, Av,...
Let A be an n × n matrix, v a column vector, and suppose {v, Av, . . . , An−1v} is linearly independent. Prove that if B is any matrix that commutes with A, then B is a polynomial in A.
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Let P(u) be the linear function mapping vector x ∈ Rn to the difference between x...
Let P(u) be the linear function mapping vector x ∈ Rn to the difference between x and the projection of xon the line L(0,u) (the line through zero with direction u.) What is the smallest and second smallest eigenvalue of P(u)?
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT