Question

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 the distribution of 1THx. Here, 1 = (1,...,1) ∈Rn is the vector of all ones.

Answer #1

**solution:-**

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.

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 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...

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