 # Linear Algebra with Applications (3rd Edition)

• Homework Help
• davidvictor
• 2

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 1 - 2 out of 2 pages.

Quiz 2 SolutionsFebruary 28, 2005Problem 1For the first part: the equation of the plane orthogonal to agiven~vR3is~x·~v= 0. Applying this formula to our problem and writing~x= (x, y, z), we getx+ 2y+ 3z= 0.For the second part, we have a formula for the reflection of~win a planeP:refP(~w) =~w-2~wwhere~wis the component of~wperpendicular toP. But this is just~w-2(~w·ˆvv,where ˆvis a unit vector normal toP.Applying this formula with ˆv=114,~w= , we getrefP(~w) =17-2-25-48Problem 2First we find the elementary row operations that reduceAtoreduced row echelon form (if possible):1012-10011---------→-2(I) + (II)*1010-1-2011---------→(II) + (III)*1010-1-200-1---------------------→-2(III) + (II)*,(III) + (I)*1000-1000-1-----------→-(II)*,-(III)*1000100011
SinceAcan be reduced to the identity matrix by elementary row operations,Ais invertible.Applying these same row operations to the 3×3 identitymatrix gives us
End of preview. Want to read all 2 pages?

Course Hero member to access this document

Term
Spring
Professor
CONSANI
Tags
Invertible matrix, elementary row operations

Unformatted text preview: matrix gives us A-1 : A-1 = -1 1 1-2 1 2 2-1-1 Problem 3 Let ~v = 1 √ 2 1 1 . Note that the plane P defined by x + z = 0 is just the plane normal to ~v . The problem demands a linear transformation T : R 3 → R 3 with kernel ~v and image P . Such a transformation T is given by T ( ~x ) = proj P ~x . To see this, note that ~v , being normal to P , is in the kernel of T . Moreover, T ( R 3 ) = P by definition. So this is the transformation we’re looking for. To find the matrix of T , write proj P ~x = ~x-~x ⊥ = ~x-( ~x · ~v ) ~v. = ( I-proj ~v ) ~x. The matrix of proj ~v is (from HW 3) B = 1 / 2 1 / 2 1 / 2 1 / 2 Thus the matrix of T is I-B , or 1 / 2-1 / 2 1-1 / 2 1 / 2 2...
View Full Document

• • • 