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Friday November 20START: 16:10DURATION: 110 minsUniversity of TorontoDepartment of MathematicsMIDTERM EXAMINATION IISOLUTIONSMAT223H1FLinear Algebra IEXAMINERS: T. Bazett, I. Biborski, N. Garcia-Fritz, S. Homayouni-Boroojeni, A. Kolpakov, H. Nuchi, S. UppalLast Name (PRINT):Given Name(s) (PRINT):Student NUMBER:Student SIGNATURE:Tutorial Group:Instructions.1. There are63possible marks to be earned in this exam. The examination booklet contains a total of 11 pages. It is yourresponsibility to ensure thatno pages are missing from your examination. Do not detach any pages from your examination.2. You may write in black ink, blue ink, or in pencil. Do not write in red ink, and ensure that your solutions are LEGIBLE.3. No aids of any kind are permitted. CALCULATORS AND OTHER ELECTRONIC DEVICES (INCLUDING PHONES)ARE NOT PERMITTED.4. Have your student card ready for inspection.5. There are no part marks for Multiple Choice (MC) questions.6. For the full answer questions, write the solutions on the question pages themselves. You may use the two blank pages atthe end for rough work.The last two pages of the examination WILL NOT BE MARKED unless youclearlyindicateotherwise on the question pages.7. For the full answer questions, show all of your work and justify your answersbut do not include extraneous information.8. Only solutions that are written in ink with no correction fluid or correction tape can be considered for re-grading.FOR MARKER USE ONLY:MCQ1Q2Q3Q4Q5Q6TOTAL1
Part I - Multiple Choice Answer Key(1): (A)(2): (B)(3): (A)(4): (C)(5): (A)1. Which of the following statements are TRUE?(i) It is possible for a linear transformation fromR2toR2to transform a parallelogram to a circle.(ii) It is possible for a linear transformation fromR2toR2to transform a parallelogram to a line segment.(iii) It is possible for a linear transformation fromR2toR2to transform a parallelogram to a triangle.(A)(ii) only(B)(i) only(C)(i) and (iii) only(D)(i) and (ii) only(E)(i), (ii), and (iii)Answer:(A) (ii) only.(i):False. Linear transformations take straight lines to straight lines (or to a single point in certain situations). Since the sidesof a parallelogram are straight lines, it follows that linear transformations cannot take a parallelogram to anything other than apolygon (which a circle is not).(ii):True. As before, linear transformations take straight lines to straight lines. However, it is possible for a linear transformationto take the two adjacent sides of a parallelogram, and map them onto the same line.For example, supposeT:R2→R2is given byT(x) =1010x. What doesTdo to the squareSwith vertices (0,0),(1,0),(0,1)and (1,1)?We find thatT00=00,T10=11,T01=00, andT11=11. We see thatTtakes theSto the linesegment from (0,0) to (1,1).1111T(iii):False. Linear transformations take parallelograms to either parallelograms, straight lines, or single points. They cannottake a parallelogram to a triangle. Suppose we have a parallelogram P:ABCDSince--→AB=--→CD, we know thatTtakes the line segmentsABandCDto parallel line segments of equal length. The same is