DOC PREVIEW
CR MATH 45 - Exam #2 Linear Algebra

This preview shows page 1-2-3-4 out of 13 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath 45 — Linear AlgebraExam #2Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: November 1, 2001 Version 1.002Essay QuestionsRead Carefully! You have the weekend to complete the exam. The exam is due, on my desk, at thebeginning of class on Monday.This exam is open notes, open b ook. You may use a calculator or computer to check your work whereappropriate. You must answer all of 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 to help with the exam. You are notallowed to read the exercises in the exam, then seek help on similar questions. Once you open the exam andread the questions, you may not seek any outside help of any kind. From the moment you open the exam,you must do everything by yourself.Place the solution to each exercise on a separate sheet of paper. On a good sheet of paper, write out(longhand) and sign the following honor pledge.I promise that all work found herein is my own. I have received no help from tutors, colleagues,or other teachers. I have honored all of the examination constraints listed in the directions.Arrange the problems in order, place these examination pages on top of your solutions, then place the honorpledge on top of the examination as a cover sheet. Staple. Good luck!Exercise 1. Let V be the vector space of all functions f from R to R with the usual definitions of additionand 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 a subspace of V .Exercise 2. Matrix[R d]=110 2 −22001−11−1000000is the reduced row echelon form of the augmented matrix [A b] for the 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 the system Ax = b.(c) Finally, without doing any further work, give the “complete” solution of the system Ax = b.Exercise 3. If v1, v2,andv3are linearly independent, use the definition of linear independence to provethat 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. Craft a Strang diagram. Place the basis foreach space in the appropriate location in your Strang diagram. Mark the dimensions of each space on yourdiagram. Mark the space in which the row space and nullspace of A dwell. Do a similar thing for the spacein which the column space of A and the left nullspace of A live.Exercise 5. Let S be the subspace of R4spanned by the vectors1−223,2−100, and7−869.3Find S⊥,theorthogonal complement of S.Exercise 6. On page 169, Figure 4.2, Strang claims that any vector x in Rncan be written as the sum of avector from the row space and the nullspace of matrix A. Let’s show how one would do this. First, considerthe matrixA =1224.(a) Find the row space and nullspace of matrix A. On a sheet of graph paper, draw both the row space andthe nullspace in the plane.(b) On the same coordinate system used in part (a) to record the row space 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 xonto the nullspace of matrix A in the same sketch. Call this vector xn. Show, using the parallelogrammethod, thatx = xr+ xn.(d) Find exact representations for xrand xn. No decimals allowed in your answer. Verify algebraically thatx = 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 line of best fit. Plot the line represented bythis equation on the coordinate system established in part (a). Note: I want to see your work. Handcalculations only.Solutions to Exercises 4Solutions to ExercisesExercise 1. Let V be the collection of all f : R → R.LetW be the collection of all f : R → R that areeven functions. Recall that a function 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 need to show two things: (1) W is closedunder addition, and (2) W is closed 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 theirdomains. 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 even function and is back in W . Therefore, Wis closed under addition.2. Let fW, CR. Then f is even and f (−x)=f(x) for all x in the 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 function and is back in W . Therefore, W isclosed under scalar multiplication.Finally, because W is closed under addition and scalar multiplication, W is a subspace of V .Exercise 1Exercise 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 vectorfor the nullspace. Column 2 can be written as a linear combinations of the pivot columns that proceed it.col2 = 1 · col1Therefore,−1 · col1 + 1 · col2 + 0 · col3 + 0 · col4 + 0 · col5 = 0and−11000is a basis vector for the nullspace. The next free column is column 4. It can be written as a linear combinationof preceding pivot columns.col4 = 2 · col1 − 1 · col3.Therefore,−2 · col1 + 0 · col2 + 1 · col3 + 1 · col4 + 0 · col5 = 0Solutions to Exercises 5and−20110is a basis vector for the nullspace. Next, the fifth column can be written as a linear combination of thepreceding pivot columns.col5 = −2 · col1 + 1 · col3.Therefore,2 · col1 + 0 · col2 − 1 · col3 + 0 · col4 + 1 · col5 = 0and20−101is a basis vector for the nullspace. ThusB =−11000−2011020−101is a basis for the nullspace. Exercise 2(b) The first two rows givex1+ x2+2x4− 2x5=2x3− x4+ x5= −1Setting all free


View Full Document

CR MATH 45 - Exam #2 Linear Algebra

Documents in this Course
Load more
Download Exam #2 Linear Algebra
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Exam #2 Linear Algebra and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Exam #2 Linear Algebra 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?