3. (4 points) Let u, v, and x be vectors in IRT. If x is orthogonal to both u an

Question

3. (4 points) Let u, v, and x be vectors in IRT. If x is orthogonal to both u and v, show that x is orthogonal to every vector in the subspace W-Span(u, u} 4. (4 points) Let W be a subspace of IR". Let W- consist of the vectors in IR" that are orthogonal 5. (4 points) Let u be a nonzero vector in IR3. Show that T: IR - R" defined by T(a) -projur 6. (4 points) Let U and V be orthogonal matrices. Show that UV is also an orthogonal matrix. to every vector in W. Show that W1 is a subspace of " is a linear transformation

Explanation / Answer

3) Subspace W is spanned by u and v. So any vector y in W can be written as linear combination of u and v.

Let y= Au + Bv

Let's take the dot product of x and y.

x.y

=x.(Au+Bv)

=Ax.u+Bx.v

Since x is perpendicular to both u and v so x.u=0 and x.v=0.

So x.y=0. So x is orthogonal to every vector in subspace W.

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.