Math 206 Study Guide for Test 2 Take-home test Apr 26 - May 3

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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 available Friday April 26 and due Friday, May 3, by 6:30 p.m.
• Schedule 90 minutes.
• Covers HWK 8 - HWK 21 and GP 8 - GP 22
• Covers Section 4.2, Chapter 5, and Sections 8.1-8.3. You might find it helpful to browse or study 4.3 but 4.3 won't be explicitly tested.
• Will include short-answer questions, computational problems, and proofs
• 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!
• Section 4.2
  • Be familiar with such language as map or transformation or function, operator, domain, codomain, image of a vector, standard matrix for a linear transformation from Rⁿ to R^m, zero transformation, identity operator, reflection operator, projection operator, rotation operator, dilation operator, contraction operator, composition of transformations.
  • Know how to find the standard matrix for reflections about coordinate axes or coordinate planes and for the reflection about y = x (in R²).
  • Know how to find the standard matrix for orthogonal projections onto coordinate axes or coordinate planes.
  • Know how to find the standard matrix for a rotation operator on R².
  • Know how to find the standard matrix for a dilation or contraction operator.
  • Know how to use matrix multiplication to find the standard matrix for a composition. And be sure to do the multiplication in the proper order.
  • Understand how matrix multiplication can be viewed as composition of functions and why/how this relates to matrix multiplication not being commutative.
  • Suggested review problems 4.3 p185: 1bc, 2d, 3, 5, 6c, 7b, 8, 9, 10, 11, 12, 17, 20, 25a, 26, 29, 31
• Section 4.3
  • Most of this material is discussed in Ch 8 in a more general context. You may find it helpful to read this section as additional background for Ch 8.
  • Note that what is a theorem about matrix transformations from Rⁿ to R^m in 4.3 becomes a definition in Ch 8. (Thm 4.3.2).
  • Suggested review problems 4.3 p198:1 any, 2 any, 3, 5 any, 7, 8
• Section 5.1. General Vector Spaces.
  • Was covered on Test 1, won't explicitly be tested here.
  • However, you should continue to be familiar with the axioms for a vector space, as well as the other properties that are established here. On this test you may use any and all of those properties/axioms.
• Section 5.2.Subspaces.
  • Some of section was also covered on Test 1 and won't explicitly be tested here.
  • You should definitely know the definition for subspace and how to prove that some subset of a vector space is or is not a subspace.
  • Know the geometric characterizations for subspaces of R² and R³.
  • Know/understand why the solution set for a homogeneous system of equations forms a subspace of the relevant Rⁿ, but the solution set for a nonhomogeneous system does not.
  • Know the notation span(S) or span{v₁, v₂, ..., v_r }. Know the related terminology. (see p217).
  • Know the condition given on p219 for when two finite sets have the same span.
  • Know what it means for a set S to span a vector space V.
  • Study Example 12 p218.
  • Suggested problems 5.2 p219: 1-5 a few, 6 any, 7-11 a few, 13, 16, 17, 19, 23, 25
• Section 5.3. Linear Independence.
  • Know (well) the two alternative characterizations for a finite set of vectors to be linearly independent (dependent). Know why these characterizations are equivalent. Be able to state these characterizations accurately and to use them accurately.
  • Be able to work with these characterizations in a variety of situations. Know how to structure a proof that a given set is independent. Know some ways to structure a proof that a given set is not independent.
  • Know some conditions/situations that will guarantee a given set is dependent.
  • Know some conditions/situations that will guarantee a given set is independent.
  • Know/understand why more than n vectors in Rⁿ will always form a dependent set.
  • Know various conditions that will guarantee a set of exactly n vectors in Rⁿ form an independent set.
  • Not covered: Wronskians and using Wronskians to establish independence.
  • Be able to prove that any set to which the zero vector belongs is necessarily dependent.
  • Be able to prove that subsets of (finite) independent sets are independent.
  • Be able to prove that (finite) sets containing dependent sets are dependent.
  • Be able to prove that a set containing exactly two vectors is dependent iff one vector is a scalar multiple of the other. Must each be a scalar multiple of the other? (answ: no; know why)
  • Given a finite set in some familiar vector space, be able to decide whether the set is independent or not.
  • Suggested problems 5.3 p229: any from 1-4, 8, 9 or 10, 11 or 12, 13, 14, 17, 23, 24
• Section 5.4. Basis and Dimension.
  • Know what it means for a (finite) set to be a basis for a vector space V. Know how to prove that two (finite) bases for the same vector space must have the same number of vectors.
  • Know what is meant by a zero-dimensional vector space.
  • Know what is meant by a finite-dimensional vector space and by the dimension of such a space.
  • Know what is meant by an infinite-dimensional vector space and know some examples of such.
  • Be familiar with the standard bases for familiar spaces. Consequently know the dimensions for these spaces.
  • Know the result called "Uniqueness of Basis Representation" (Thm 5.4.1 p233) and be able to prove it.
  • Know the terminology and notation for coordinates relative to a basis S and coordinate vector relative to a basis S. Note that the order in which the basis vectors are listed matters here.
  • Know Theorem 5.4.2(a) : in a space that has an n-vector basis, every set with more than n vectors is linearly dependent. You should understand the proof, but you will not be asked to do the proof. Know how to use this result to prove that two (finite) bases for the same vector space must have the same number of vectors.
  • Know Theorem 5.4.2(b) and the proof given for it in class.
  • Know how to find a basis for and the dimension of the solution space for a homogeneous system. (Exple 10 p239).
  • Know the Plus/Minus Theorem and how to prove it.
  • Know Theorem 5.4.5: in an n-dimensional vector space, any n-vector set that is independent or that spans the space will necessarily be a basis. Know how to prove it and how to use it. This theorem is frequently used in establishing that some particular set is a basis. Instead of showing both spanning and independence, if the set has the correct number of vectors then we only have to show spanning or independence.
  • Be able to prove, for a finite-dimensional space, that a (finite) spanning set always contains a basis and a (finite) independent set can always be enlarged to a basis. (Thm 5.4.6)
  • Understand the connection between bases for a subspace and bases for the whole space. Given a subspace of a finite-dimensional space, any basis for the subspace can be enlarged to a basis for the whole space. However, many bases for the whole space will contain no basis for the subspace. In fact, many bases for the whole space will contain no vectors at all from the subspace. Be able to provide examples.
  • Know the theorem about subspaces of finite dimensional spaces being finite dimensional (of dimension no larger than that of the whole space).
  • Suggested review problems 5.4 p243: 1, some from 2-4, 5, some from 7-10, some from 11-16, 17, 19, 22, 23, perhaps 26, 31a, 32, 33, 34, 35
• Section 5.5. Row Space, Column Space, and Nullspace. Rank and Nullity.
  • Know what is meant by the row space of a matrix and the column space of a matrix.
  • Know that the dimension of the row space is the same as the dimension of the column space and know some reason.
  • Know what is meant by the rank of a matrix and know how to find it.
  • Know how to find a basis for the row space, for the column space. Know how to use these methods to find a basis for the subspace of Rⁿ spanned by some given (finite) set of vectors in Rⁿ.
  • Given a matrix A, know how to choose from the columns of A a basis for the column space of A. Know how to write the other columns in terms of the chosen basis columns.Know how to use this to select, from a (finite) set S of vectors in Rⁿ a subset of S forming a basis for the span of S.
  • Know what is meant by the nullspace of a matrix and the nullity of a matrix. Know how to find a basis for the nullspace. Know how to find the nullity.
  • Know what effect elementary row operations have (or do not have) on the row space, the column space, the nullspace of a matrix. Know why or how to prove.
  • Know the dimension theorem for a matrix A. This is the theorem that connects the rank of A, the nullity of A, and the number of columns of A. Understand why it's true.
  • Know and understand the connection between the column space of A and the solutions to the system Ax=b.
  • Be conversant with the various equivalent conditions given in the Long Theorem p268. You need not memorize the list but you should be familiar with it. Understand as much as possible how these various conditions are related.
  • Suggested problems 5.5 p257: 2c, 3 any, 6 any, corresponding ones from 8-10, 11, 12, 13, 15
  • Suggested review problems 5.6 p269: 2, 3, 4, 5, 6, 7, 11, 15, 17, 18, 19
• Chapter 5 Supplementary Exercises p271: 3, 5, 6, 10, 11
• Section 8.1. General Linear Transformations
  • Know what is meant by a linear transformation from one vector space to another.
  • Be able to decide whether a given function F:V --> W is a linear transformation or not.
  • Note Exple 7. It essentially says that taking coordinates relative to a particular basis defines a linear transformation.
  • Be familiar with the properties given in Thm 8.1.1. Be able to prove them.
  • Understand how a linear transformation on a finite-dimensional vector space is completely determined by what it does to the vectors in a given basis for the space.
  • Note that the composition of two linear transformations (when defined) is also a linear transformation. Understand the proof.
  • Know what is meant by an identity operator and a zero transformation.
  • Suggested problems 8.1 p373: 3-10, 12, 13 or 14 or 15, 16, 17, 19, , 29, 30, 33
• Section 8.2. Kernel and Range.
  • Know/understand definitions for kernel of a linear transformation, range of a linear transformation.
  • Know how to decide whether a given vector is in the kernel.
  • Know how to test whether a given vector is in the kernel.
  • Be able to prove that kernel of a linear transformation is a subspace of the domain.
  • Be able to prove that range of a linear transformation is a subspace of the codomain.
  • Know and, for the case where the kernel is neither 0-dimensional nor the whole domain space, know how to prove the Dimension Theorem for linear transformations.
  • Suggested problems 8.2 p380: 3, 4, 5, 6, 7b, 8b, 9b, any from 10-13, 14ab, 15, 16, 17, 18, 22, 24, 25, 27
• Section 8.3. Inverse Linear Transformations.
  • Know what it means for a transformation to be one-to-one.
  • For one-to-one transformations, know what is meant by the inverse transformation.
  • Know how the condition "one-to-one", for linear transformations, relates to the dimension of the kernel. Know how to prove the connections. (Thms 8.3.1 & 8.3.2).
  • For one-to-one transformations defined as matrix multiplications, know how to find the inverse transformation.
  • Know Thm 8.3.3. (but you need not know the proof). The theorem says that a composition of two one-to-one linear transformations is also one-to-one and that the inverse of such a composition is the composition of the inverses, but in reverse order.
  • Review problems 8.3 p388: 1, 2 or 3 or 4, 6, 7, 8

Questions?
Send me e-mail or post in the Q&A conference in our FirstClass conference. 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: April 23, 2002
• Expires: June 30, 2002