Consider the linear mappingD4:P10(t)→P10(t), whereD4denotes the fourth derivative,d4f/dt4.Find a basis for and thedimension of(a) The image ofD4.(b) The kernel ofD4.I.6. SupposeS1,S2, andS3are bases ofV. LetPandQbe the change-of-basis matrices, respectively,fromS1toS2and fromS2toS3. Prove thatPQis the change-of-basis matrix fromS1toS3.

Homework 3: Vector and Inner Product Spaces
Page 3/4
I.7. SupposeS={u1, u2}is a basis ofVand thatT:V→Vis defined byT(u1) =-u1+ 2u2andT(u2) = 2u1+u2.SupposeS0={w1, w2}is also a basis ofVfor whichw1=u1+u2andw2= 2u1+ 3u2.(a) Find the matricesAandBrepresentingTrelative to the basesSandS0, respectively.(b) Find the matrixPsuch thatB=P-1AP.I.8. Explain why the following statement is nonsense.Two linear operatorsFandGon a vector spaceVare said to be similar if there exists an invertiblelinear operatorTonVsuch thatG=T-1◦F◦T. IfFis similar to a diagonal matrix, then anysimilar matrixGis also similar to a diagonal matrix.I.9. LetR2×2be endowed with the inner producthA, Bi= tr(BTA). Find an orthogonal basis for theorthogonal complement of(a) diagonal matrices.(b) symmetric matrices.I.10. Problem LA3.1.44.Hard Problems (40 points)H.1. SupposeVis a finite-dimensional vector space and suppose thatTis a linear operator onVsuchthat rank(T2) = rank(T). Show that Ker(T)∩Im(T) ={0}.H.2. LetF:V→Ube linear and letWbe a subspace ofV.The restriction ofFtoWis the mapF|W:W→Udefined byF|W(v) =F(v) for everyv∈W. Prove the following(a)F|Wis linear.(b) Ker(F|W) = (Ker(F))∩W.(c) Im(F|W) =F(W).Read the article, “An Introduction to Robust Codes over Finite Fields” by S. Engelberg and O. Keren(SIAM Review, volume 55, pp. 751–763, 2013). We will focus on§§1–8.H.3. The linear, systematic codes discussed add redundancy to data that helps the receiver determine ifan error was introduced during transmission, whether malicious or not. This redundancy is added