Understanding Linear Transformations and Matrices in MATH 220
School
Cumberland Valley Hs**We aren't endorsed by this school
Course
MATH 220
Subject
Mathematics
Date
Dec 12, 2024
Pages
16
Uploaded by MegaBraveryToad20
Midterm II10/26/2023MATH 220MatricesName:PRINTyour nameCLEARLYas it appears in CANVAS and LionPath.PSU Email ID:@psu.eduSection Instructor:Section number:INSTRUCTIONSWrite your initials on the bottom of each page in the indicated space.Failure to do this may result in a 5 point deduction.Check that your exam contains 13 questions numbered sequentially on 16pages.Answer all your questions in your test booklet, show all of your work, writelegibly, and write your final answer in the box or space provided.The use of a calculator, cell phone, smartwatch, or any other electronic deviceis not permitted during this examination.The use of scrap paper or notes of any kind is not permitted during thisexamination.Good luck!Page 1 of 16Initials:
You can use this page foroverflowscratch work.Please label which problem(s) it pertains to.Only use this page if you run out of space elsewhere.Page 2 of 16Initials:
1. (4 points) Provide a geometric description of how the linear transformationTacts on anyvectorx=xyinR2.(a)T(x) =0110x→Treflectsxover the origin.→Treflectsxover the liney=x.→Trotatesxω4radians clockwise.→Trotatesxω4radians counterclockwise.(b)T(x) =1000x→Treflectsxover thex-axis.→Treflectsxover they-axis.→Tprojectsxonto thex-axis.→Tprojectsxonto they-axis.(c)T(x) =1002x→Tstretchesxhorizontally by a factor of 2.→Tstretchesxvertically by a factor of 2.→Tdoubles the length ofx.→Thalves the length ofx.(d)T(x) =100↑1x→Treflectsxover thex-axis.→Treflectsxover they-axis.→Tprojectsxonto thex-axis.→Tprojectsxonto they-axis.Page 3 of 16Initials:[x]⑧[:]⑧··[i]
2. (6 points) The formula for a linear transformationTis given below. Find the standard matrixAofT. Then determine if the linear transformation is one-to-one, onto, both one-to-one andonto, or neither.(a)T:R2↓R2defined byTxy=x↑yA=Select one: The linear transformationTis→One-to-onebut not onto→Ontobut not one-to-one→Both one-to-one and onto→Neitherone-to-one nor onto(b)T:R3↓R2defined byTxyz=x+y↑z2y+zA=Select one: The linear transformationTis→One-to-onebut not onto→Ontobut not one-to-one→Both one-to-one and onto→Neitherone-to-one nor ontoPage 4 of 16Initials:[i][te7not1-1&ontovBo]]⑨
3. (8 points) SupposeT:R2↓R2is alineartransformation such thatT10=23andT1↑3=42.FindT1↑9.T1↑9=Page 5 of 16Initials:[6-s(c]=(wilsesCi15]-z(j]+3(2)=[i]12-4+6-6
4. (15 points) LetA=1302,B=↑115↑2,C=30↑2041,andD=200111002.Perform each computation below or state that the operation is undefined.(a)A+BA+B=(b)CDCD=Page 6 of 16Initials:2X33x3&97pay a
LetA=1302,B=↑115↑2,C=30↑2041,andD=200111002.Perform each computation below or state that the operation is undefined.(c)DCDC=(d)CTCT=(e)B2B2=Page 7 of 16Initials:[ii]3x32x31)][]und
5. (6 points) Determine if each matrix below is invertible or singular. Justify your response.(a)100230456→Invertible→SingularJustification:(b)123050143→Invertible→SingularJustification:(c)230460110→Invertible→SingularJustification:Page 8 of 16Initials:dut=18⑨def=01(1s)-2(0)+3)-5)=0⑨det=02(0)-3(0)+0⑧det=0
6. (8 points) LetA=103216033.Ais an invertible matrix. FindA→1.A→1=Page 9 of 16Initials:iR
7. (6 points) Define a linear transformationT:R2↓R2byTxy=2x+ 3y4x+ 5y.(a) Find the standard matrixAofT.A=(b) Find the inverseA→1of the standard matrixA.A→1=(c) Find a formula for the inverse transformationT→1.Write your formula in the formT→1xy=bwherebis a vector inR2.T→1xy=Page 10 of 16Initials:(3]#[]=(i)[x
8. (8 points) Find an LU factorization ofA=13↑5↑1↑5842↑5.L=U=Page 11 of 16Initials:
9. (12 points) LetA=3163andb=2↑3.(a)A→1=1↑1/3↑21.Use A→1to solve the linear systemAx=b.x=(b) An LU factorization ofAisA=LU=10213101.Use this LU factorizationto solve the linear systemAx=b=2↑3.x=Page 12 of 16Initials:(x2X)Ax=DAb[ii](=]=(2)
10. (8 points) Calculate the determinant ofA=1↑4027↑2210↑201↑1005usingcofactor expansion.det(A) =Page 13 of 16Initials:-OHe(t)-c)-10+(1)+2(2)20
11. (5 points) LetAandBbe 2↔2 matrices with det(A) = 3 and det(B) =↑1.Find thedeterminant of each matrix given below.(a) 2Adet(2A) =(b)A→1det (A→1) =(c)BTdet(BT=(d)ABdet (AB) =(e)A3det (A3) =Page 14 of 16Initials:2.3=12t3-I-327
12. (6 points) Determine if each subset ofR2given below is asubspaceofR2. Justify your answer.(a)xyx= 1→This setisa subspace ofR2→This setis nota subspace ofR2Justification:(b)xyx= 0→This setisa subspace ofR2→This setis nota subspace ofR2Justification:Page 15 of 16Initials:⑨[07X⑧[8]vGr(ig)
13. (8 points) LetA=13↑20226↑5↑2426084. The reduced echelon form ofAis130420012000000.(a) Find a basis for ColA.Basis for ColA:(b) Find a basis for NulA.Basis for NulA:Page 16 of 16Initials:-0o(i7(]X,=-3xz-4xy-2x5Xz==EXY~=x()+m+x[