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MATH 3260 Exam 2: Linear Transformations and Vector Spaces, Exams of Linear Algebra

Kennesaw State University (KSU)Linear Algebra

The instructions and questions for exam 2 of math 3260, a linear algebra course. Topics covered include vector space definitions, linear independence, transformations, and their properties. Students are required to write complete sentences and use correct notation in their answers.

Typology: Exams

2010/2011

Uploaded on 06/03/2011

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MATH 3260 Exam 2 (Version 1)
June 23, 2008
S. F. Ellermeyer Name
Instructions. Remember to include all important details of your work. You will not get full
credit (or perhaps even any partial credit) if I see gaps in your reasoning. Also, use correct
notation and write in complete sentences where appropriate. You may use a calculator on
this exam but you may not use any books or notes.
1. (De…nitions) Use complete sentences to write the following de…nitions.
(a) Let fv1;v2;:::;vngbe a set of vectors in Vm. What do we mean when we say
that this set of vectors is linearly independent?
(b) Suppose that T:Vn! Vmis a transformation. Given any vector x2Vn, what
is meant by the image of xunder the action of T? What is meant by the range
of T?
(c) What do we mean when we say that a transformation T:Vn! Vmis a linear
transformation?
(d) What do we mean by the identity transformation T:Vn! Vm?
(e) What do we mean when we say that a transformation T:Vn! Vmis onto Vm?
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MATH 3260 Exam 2 (Version 1) June 23, 2008 S. F. Ellermeyer Name

Instructions. Remember to include all important details of your work. You will not get full credit (or perhaps even any partial credit) if I see gaps in your reasoning. Also, use correct notation and write in complete sentences where appropriate. You may use a calculator on this exam but you may not use any books or notes.

  1. (DeÖnitions) Use complete sentences to write the following deÖnitions.

(a) Let fv 1 ; v 2 ; : : : ; vng be a set of vectors in Vm. What do we mean when we say that this set of vectors is linearly independent? (b) Suppose that T : Vn ! Vm is a transformation. Given any vector x 2 Vn, what is meant by the image of x under the action of T? What is meant by the range of T? (c) What do we mean when we say that a transformation T : Vn ! Vm is a linear transformation? (d) What do we mean by the identity transformation T : Vn ! Vm? (e) What do we mean when we say that a transformation T : Vn ! Vm is onto Vm?

  1. Determine whether or not the three vectors that comprise the columns of the matrix

A =

form a linearly independent set. You must justify your answer (writing in complete sentences).

  1. Let T : V 3 ! V 2 be the linear transformation deÖned by

T

x y z

A =

x + 4y 5 z 3 x 7 y + 4z

(a) Find the standard matrix of this linear transformation. (b) Is T onto V 2? Explain why or why not. (No explanation = no credit.) (c) Is T oneñtoñone? Explain why or why not. (No explanation = no credit.)

  1. This year, Club A has 4000 members and Club B has 6000 members. Each year, 5% of Club Aís members quit Club A and join Club B. Also, each year, 5% of Club Bís members quit Club B and join Club A and 2% of Club Bís members quit club B and donít join any other club. Set up a di§erence equation that describes this situation. (Make sure to state what all variables you are using stand for.) Then, by calculator computation, determine the number of members that each club will have ten years from now. (You do not have to write down every intermediate computation. But do show at least the Örst couple of iterations. Round your answers to the nearest whole number.)