Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath 45 — Linear AlgebraExam #2Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: November 2, 2001 Version 1.002Essay QuestionsRead Carefully! You have the weekend to complete the e xam. Theexam is due, on my desk, at the beginning of class on Monday.This exam is open notes, open book. You may use a calculator orcomputer to check your work where appropriate. You must answer allof the exercises on your own. You are not allowed to work in groups onthe exam. You are not allowed to enlist the aid of a tutor or friend tohelp with the exam. You are not allowed to read the exercises in theexam, then seek help on similar questions. Once you open the examand read the questions, you may not seek any outside help of anykind. From the moment you open the exam, you must do everythingby yourself.Place the solution to each exercise on a separate sheet of paper.On a goo d sheet of paper, write out (longhand) and sign the followinghonor pledge.I promise that all work found herein is my own. I havereceived no help from tutors, colleagues, or other teachers.I have honored all of the examination constraints listed in3the directions.Arrange the problems in order, place these examination pages on topof your solutions, then place the honor pledge on top of the examina-tion as a cover sheet. Staple. Good luck!Exercise 1. Let V be the vector space of all functions f from R toR with the usual definitions of addition and scalar multiplication.(f + g)(x)=f(x)+g(x)(cf)(x)=cf(x)Let W be the set of all even functions in V . Prove that W is asubspace of V .4Exercise 2. Matrix[R d]=110 2 −22001−11−1000000is the reduced row echelon form of the augmented matrix [A b]forthe system Ax = b.(a) Without doing any work, give a basis for the nullspace of A.(b) Again, without doing any work, give a particular solution of thesystem Ax = b.(c) Finally, without doing any further work, give the “complete” so-lution of the system Ax = b.5Exercise 3. If v1, v2,andv3are linearly independent, use the defini-tion of linear independence to prove that v1, v1+v2,andv1+v2+v3are linearly independent.Exercise 4. Consider matrixR =10 101−100 000 000 0.Find a basis for each of the four fundamental subspaces of R.Crafta Strang diagram. Place the basis for each space in the appropriatelocation in your Strang diagram. Mark the dimensions of each spaceon your diagram. Mark the space in which the row space and nullspaceof A dwell. Do a similar thing for the space in which the column spaceof A and the left nullspace of A live.6Exercise 5. Let S be the subspace of R4spanned by the vectors1−223,2−100, and7−869.Find S⊥,theorthogonal complement of S.7Exercise 6. On page 169, Figure 4.2, Strang claims that any vectorx in Rncan be written as the sum of a vector from the row space andthe nullspace of matrix A. Let’s show how one would do this. First,consider the matrixA =1224.(a) Find the row space and nullspace of matrix A. On a sheet of graphpaper, draw both the row space and the nullspace in the plane.(b) On the same coordinate system used in part (a) to record the rowspace and nullspace of matrix A, sketch the vectorx =08.(c) Project x onto the row space of A in the sketch made in part (a).Call this vector xr. Next, project x onto the nullspace of ma-trix A in the same sketch. Call this vector xn. Show, using theparallelogram method, thatx = xr+ xn.8(d) Find exact representations for xrand xn. No decimals allowed inyour answer. Verify algebraically that x = xr+ xn.Exercise 7. Consider the following (x, y) pairs.(1, 0), (3, 0), (5, 6)(a) Plot the given points on a sheet of graph paper.(b) Using the method of least squares, find an equation for the lineof best fit. Plot the line represented by this equation on the co-ordinate system established in part (a). Note: I want to see yourwork. Hand calculations only.Solutions to Exercises 9Solutions to ExercisesExercise 1. Let V be the collection of all f : R → R.LetW bethe collection of all f : R → R that are even functions. Recall that afunction is even if and only iff(−x)=f(x)for all x in one domain of f. To show that W is a subspace, we needto show two things: (1) W is closed under addition, and (2) W isclosed under scalar multiplication.1. Let f, gW. Then f and g are both even and f (−x)=f (x)andg(−x)=g(x) for all x in their domains. Now,(f + g)(−x)=f(−x)+g(−x)= f (x)+g(x)(f + g)(x)for all x in the domain of f + g. Therefore, f + g is an evenfunction and is back in W . Therefore, W is closed under addi-tion.Solutions to Exercises 102. Let fW, CR. Then f is even and f (−x)=f(x) for all x inthe domain of f.Now,(cf)(−x)=cf(−x)= cf (x)=(cf)(x)for all x in the domain of cf. Therefore, cf is an even func-tion and is back in W . Therefore, W is closed under scalarmultiplication.Finally, because W is closed under addition and scalar multipli-cation, W is a subspace of V .Exercise 1Solutions to Exercises 11Exercise 2(a) If[R d]=110 2 −22001−11−1000000,then we have pivots in columns one and three. Each non pivot column(free column) produces a basis vector for the nullspace. Column 2 canbe written as a linear combinations of the pivot columns that proceedit.col2 = 1 · col1Therefore,−1 · col1 + 1 · col2 + 0 · col3 + 0 · col4 + 0 · col5 = 0and−11000Solutions to Exercises 12is a basis vector for the nullspace. The next free column is column 4.It can be written as a linear combination of preceding pivot columns.col4 = 2 · col1 − 1 · col3.Therefore,−2 · col1 + 0 · col2 + 1 · col3 + 1 · col4 + 0 · col5 = 0and−20110is a basis vector for the nullspace. Next, the fifth column can bewritten as a linear combination of the preceding pivot columns.col5 = −2 · col1 + 1 · col3.Therefore,2 · col1 + 0 · col2 − 1 · col3 + 0 · col4 + 1 · col5 = 0Solutions to Exercises 13and20−101is a basis vector for the nullspace. ThusB =−11000−2011020−101is a basis for the nullspace. Solutions to