Question

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.

Answer #1

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 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 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 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 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 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, . . . , 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〉 ∈ 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 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)?

Let B = [ aij ] 20×17 be a matrix with real entries. Let x be in
R 17 , c be in R 20, and 0 be the vector with all zero entries.
Show that each of the following statements implies the other.
(a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b)
If Bx = c has a solution for some vector c in R 20, then the
solution is unique.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 20 minutes ago

asked 22 minutes ago

asked 27 minutes ago

asked 44 minutes ago

asked 48 minutes ago

asked 48 minutes ago

asked 52 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago