Applied Math 245 Midterm Exam: Winter 2007
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1 Applied Math 45 Midterm Exam: Winter 7 Answerthequestionsinthespacesprovidedonthequestionsheets.Ifyourunoutofroomforan answercontinueonthebackofthepage.nonotes,books,orotheraidsmaybeusedontheexam. Student Id: Answer Key. ( points). ( points). ( points) 4. ( points) 5. ( points) 6. ( points) 7. ( points) 8. ( points) 9. ( points). ( points) Total( points) Pageof7
2 Applied Math 45 Midterm Exam: Winter 7.( points) Determine whether each of the following statements is true or false. No justification is required. (a) A linear system with fewer unknowns than equations must have infinitely many solutions or none. Solution: False;thesystemx,y,x+y5hasauniquesolution. (b)ifaandbarematricesofthesamesize,theformularank(a+b)rank(a)+rank(b)musthold. Solution: False;letABI forexample. [ ] [ ] x (y+) (y ) (c)thefunctiont y (x ) (x+) represents a linear transformation. [ ] [ ] [ ] [ ] x 4y 4 x Solution: True; simplify to see that T. y x y (d) The matrix [ ] k is invertible for all real numbers k. 5 k 6 Solution: True;det(A)k 6k+,whichhasnorealsolution. (e) The vectors,, formabasisforr. Solution: True; by inspection the vectors are linearly independent. Pageof7
![(b)ifaandbarematricesofthesamesize,theformularank(a+b)rank(a)+rank(b)musthold. Solution: False;letABI forexample. [ ] [ ] x (y+) (y ) (c)thefunctiont y (x ) (x+) represents a linear transformation.](/docs-images/50/14930500/images/page_2.jpg "[ ] [ ] [ ] [ ] x 4y 4 x Solution: True; simplify to see that T. y x y (d) The matrix [ ] k is invertible for all real numbers k. 5 k 6 Solution: True;det(A)k 6k+,whichhasnorealsolution.")
3 Applied Math 45 Midterm Exam: Winter 7.(points) Findallsolutionstothesetofequations x + x b x + x b. Solution: Byinspection,weseethatthesystemisconsistentonlyifb b..(points) FindallvectorsinR 4 thatareperpendiculartothethreevectors,, Solution: Weneedtofindavector xsuchthat v x v x v x.wecanrepresentthisasthe systema x,wheretherowsofaarethethreevectorsgiven.wecaneasilycompute:.5 rref Lettingx 4 t,weseethatanyvectoroftheform xt.5.5 isasolution.. Pageof7
4 Applied Math 45 Midterm Exam: Winter 7 4.(points) LetB [ ] and(ab) 5 Solution: A(AB)B ((AB) ) B [ ].FindA. 5 [ ] [ ] 5 5 [ ][ ] 5 5 [ ] ( points) Describe each of the following linear transformations as a well-known geometric transformation combined with a scaling. Give the scaling factor in each case. [ ] (a) [ ] u u u Solution: Thismatrixhastheformofaprojection: k [ ] u u u u u u u u u [ ] [ /k /k,whereu /k /k +u k + k.thus,kand u u u [ ], or equivalently ] [ ]. (b) [ ] Solution: This matrix has the formofaverticalshear: k [ ] [ ] /k.thus,kandc c /k /k. [ ] c [ ], or equivalently (c) [ ] 4 4 Solution: Thismatrixhastheformofareflection:k [ ] [ a b /k 4/k b a 4/k /k [ ] a b b a ],wherea +b ( k ) + ( 4 k ).Thus,k5. [ ] 4, or equivalently 4 Page4of7
![[ ] (a) [ ] u u u Solution: Thismatrixhastheformofaprojection: k [ ] u u u u u u u u u [ ] [ /k /k,whereu /k /k +u k + k.thus,kand u u u [ ], or equivalently ] [ ].](/docs-images/50/14930500/images/page_4.jpg "(b) [ ] Solution: This matrix has the formofaverticalshear: k [ ] [ ] /k.thus,kandc c /k /k.")
5 Applied Math 45 Midterm Exam: Winter 7 6.(points) Thecoloroflightcanberepresentedasavector R G, B whereramountofred,gamountofgreen,andbamountofblue.thehumaneyeandthebrain transform the incoming signal into the signal I L, S where intensity I R+G+B long-wavesignal L R G short-wavesignal S B R+G. R I (a) Find the matrix P representing the transformation from G to L. B S Solution: P (b)considerapairofyellowsunglassesforwatersportsthatcutsoutallbluelightandpassesallredand green light. Find the matrix A that represents the transformation incoming light undergoes as it passes through the sunglasses. Solution: A (c) Find the matrix for the composite transformation that light undergoes as it first passes through the sunglasses and then the eye. Solution: PA Page5of7
6 Applied Math 45 Midterm Exam: Winter (points) FindvectorsthatspanthekernelofA. Solution: Reducing the matrix to reduced row-echelon form: 4 Lettingx t,weseethatanyvectoroftheformt isasolution.thus,ker(a)span. 8.(points) ExpressthekernelofA astheimageofanothermatrixb. Solution: Thismatrixisalreadyinreducedrow-echelonformandhasthreefreevariables.Lettingx r, x 5 sandx 6 t,weseethatsolutionsareoftheform: r s r s s r +s +t span,, s t Sincetheimageofamatrixissimplythespanofitscolumns,wehave: B Page6of7
7 Applied Math 45 Midterm Exam: Winter 7 9.(points) FindabasisoftheimageofA 5. 7 Solution: The first two columns are linearly independent, since the second is not a scalar multiple of the first.thethirdcolumnisnotalinearcombinationofthefirsttwo,whichcanbeseenfrom: rref 5 7 Therefore,abasisfortheimageis,, 5. 7.(points) Verifythattheimageofann mmatrixaisasubspaceofr n. Solution: We need to prove that im(a) meets the three necessary conditions:.a foranya.therefore, isinim(a)..let x, ybeinim(a).then,bydefinitionoftheimage,thereexistvectors vand wsuchthata v x anda w y.weseethat x+ ya v+a wa( v+ w).therefore, x+ yisinim(a)..let xbeinim(a).then,bydefinitionoftheimage,thereexistsavector vsuchthata v x.we seethatk xka va(k v).therefore,k xisinim(a). Page7of7
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