Math 206 Study Guide for Test 1 Take-home test Mar 11-14

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• Closed-book, except that you will be allowed one page of 8 1/2'' by 11'' paper to use as a crib sheet.
• To be handed out Monday, March 11 and due Thursday, March 14 at class time
• Schedule 2 hours.
• Covers HWK 1 - HWK 11 and GP 1-GP4, GP6, GP7 Prob 26
• Covers Chapters 1-2, Section 5.1, and pp 211-215 of 5.2.
• You should also be familiar with the material in 3.1, 3.2, 4.1, and definition of dot product from 3.3.
• Expect some computational problems
• Expect some short-answer questions
• Expect some proofs
• Specific topics and suggested review problems are listed below. I've listed lots of problems, in case there's a particular topic you especially want to practice. Don't try to do all the suggested problems. Remember that some problems make good review or study problems but not good test problems, and vice versa.
• See the Quiz 1 Study Guide for list of items from 1.1-1.6
• 1.1 p6: 3d, 7, 12
• 1.2 p19: 3, 4acd, 5ac, some parts from 6-11, 12, 18, 22, 25 or 26, 31, 32
• 1.3 p33: perhaps some parts from 1-6, 7bf, 8b, 9, 11, 12, 18, 19, 22, 23, 28, 29, 30
• 1.4 p47: perhaps some parts from 1-8, 9b, 12, 13, 14, 15, 16, 17, 21, 22, 29, 31, 32, 35, 36
• 1.5 p56: perhaps some parts from 1-4, some parts from 6-7, 8a, 9, 11, 12, 14, 15, 16, 18, 19, 20
• More from Section 1.6
  • using inverse matrix to solve system
  • using inverse matrix to solve several systems with common coefficient matrix
  • know Thm 1.6.3 (and understand the proof)
  • know Thm 1.6.5
  • know how to determine values of b for which Ax=b is consistent (Exples 3,4 p63)
  • 1.6 p64: 9, 15, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30
• Section 1.7
  • know what it means for a square matrix to be diagonal, upper triangular, lower triangular
  • know how to spot whether diagonal matrix is invertible or not and, if so, how to find inverse
  • know how to find powers of a diagonal matrix
  • know how to multiply a matrix by a diagonal matrix
  • know definition of symmetric matrix
  • 1.7 p71: some parts from 1-5, 15a, 22, 30
• Ch 1 Supplementary Exercises p74: 1, 3, 5, 14a, 15, 16, 20, 21
• Section 2.1
  • definition of elementary product
  • definition of signed elementary product
  • how to count inversions
  • definition of determinant as sum of signed elementary products
  • 2.1 p87: 1c, 2c, 13b, 17b, 21, 22, 23
• Sections 2.2 and 2.3
  • know several different conditions that give det A = 0
  • know connections between det A, det A^T, det A^-1, det kA
  • know connections between det A, det B, and det AB
  • det(A+B) is in general not the same as det A + det B
  • know connections between determinants and elementary row operations
  • know conditions on A,B, and C that will give det C = det A + det B
  • know connection between determinants and invertibility
  • know how to quickly find determinant for diagonal or triangular matrix (and understand why the method works)
  • be familiar with the Long Theorem
  • understand the proofs for various equivalences in the Long Theorem
  • 2.2 p94: perhaps some from 1-3, 8, 11, 12, 14, 16, 17
  • 2.3 p102: perhaps some from 1-4, 5, 6, 9, 12, 16, 18, 20, 21, 22, 23
• Section 2.4
  • definition for minor M_ij
  • definition for cofactor C_ij
  • evaluating determinants by using cofactor expansion
  • Cramer's Rule (was not included on Quiz 2 but is included here)
  • not included: formula for inverse matrix
  • not included: definition of adjoint
  • 2.4 p112: perhaps some from 1-3, 7, 9, 23, 35b
• Chapter 2 Supplementary Exercises p115: 2, 3, 6
• Section 5.1
  • Be familiar with the axioms for a vector space, but you need not memorize them as a list
  • Be able to decide whether suggested operations of addition and scalar multiplication satisfy the vector-space axioms or not
  • Be familiar with the following vector spaces (all with the usual operations):
    • Rⁿ
    • the set M_mn or M_m,n of all m x n matrices
    • the set F(-infinity, infinity) of all real-valued functions defined on the entire real line.
  • Be familiar with the examples of vector spaces that we examined in class (the moon example, and the example where the zero vector is (1,1)).
  • Be familiar with
    • properties of scalar multiplication given in Theorem 5.1.1 p208
    • vector-space cancellation law for addition (know how to prove)
    • zero vector is unique (know how to prove)
    • negatives are unique (know how to prove)
  • 5.1 p209: any from 1-17, 21, 22
• Section 5.2
  • Know what is meant by a subspace of a vector space
  • Be able to decide whether a given set is a subspace or not
  • Know how to prove some subset is a subspace
  • Know how to prove some subset is not a subspace
  • Be familiar with the geometric description for the various subspaces of R² and R³.
    • Possible subspaces of R² are zero subspace, lines through the origin, and whole space.
    • Possible subspaces of R³ are zero subspace, lines through the origin, planes through the origin, and whole space
    • Given some other subset, be able to show it is not a subspace. (See Exple 3.)
  • A vector space with at least two vectors has at least two subspaces.
    • What are they?
    • What examples can you think of where those are the only two subspaces?
  • Be familiar with the example from Thm 5.2.2: solution set for a homogeneous linear system in n unknowns is a subspace of Rⁿ. Given a nonhomogeneous system, be able to show that its solution set is not be a subspace.
  • 5.2 p219: any from 1-5, 6, 17, 20, 23ab

Questions?
Send me e-mail or post in our Q&A subconference on FirstClass. When applicable, give page or section number, give problem number, and be specific.

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 4, 2001
• Last Modified: March 9, 2002
• Expires: June 30, 2002