General Information
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The exam will be held on Thursday, November 1, from 5:30pm to 7:00pm, in Smith Labs 154.
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Plan to arrive a few minutes early to allow time to distribute the exams.
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The exam will cover material from sections 2.1, 2.2, 2.3, and 2.4 in the text.
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Cell-phones should be turned OFF for the duration of the exam.
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You may use a non-graphing, scientific calculator during the exam. No other calculator or electronic device may be used during the exam.
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This is a closed-book exam. No books or notes may be used during the exam.
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You will be expected to show all of your work. A correct answer with insufficient justification
may not receive full credit.
Topics
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Sums and Intersections of Subspaces Know the definition of the sum and intersection of subspaces of a vector space, and how their dimensions are related.
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Linear Transformations. Know the definition of a linear transformation, and be familiar with the examples of linear transformations covered in class and in the examples in the text. Know how to find the matrix for a linear transformation with respect to bases for the vector spaces it maps between.
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Kernel and Image of a Linear Transformation Know the definitions of the kernel and image of a linear transformation, and be able to find them for a given linear transformation. Know how their dimensions are related.
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Injectivity and Surjectivity Know the definitions of injective and surjective, and be able to determine whether a given linear transformation is injective and/or surjective.
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Inhomogeneous Equations Know how the solvability and solution sets of an inhomogeneous equation T(x)=b are related to the image and kernel of T
Important Definitions/Theorems/Axioms
I will expect you to know and be able to use all of the definitions, examples and theorems below to prove results similar to those on the homework assignments. You should know precise statements of the definitions and theorems in bold.
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Theorem 1.6.18
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Definition 2.1.1
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Proposition 2.1.14
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Definition 2.2.6
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Examples 2.2.8 and 2.2.9
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Definition 2.2.10
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Proposition 2.2.15
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Definition 2.3.1
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Proposition 2.3.2
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Definition 2.3.10
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Proposition 2.3.11
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Proposition 2.3.12
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Theorem 2.3.17
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Proposition 2.4.2 and Corollaries 2.4.3, 2.4.4 and 2.4.5
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Proposition 2.4.7 and Corollaries 2.4.8 and 2.4.9
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Proposition 2.4.10
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Proposition 2.4.11 and Corollary 2.4.15
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Proposition 2.4.16 and Corollary 2.4.17
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Proposition 2.4.19 and Corollary 2.4.20
Preparing for the Exam
The exam will consist of questions similar to those on the homework assignments. Be sure you are familiar with how to solve these types of problems. I would also recommend using the following exercises for practice:
