M 340L CS Homew ork Set 6 Solutions
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1 1. Suppose P is ivertible ad M 34L CS Homew ork Set 6 Solutios A PBP 1. Solve for B i terms of P ad A. Sice A PBP 1, w e have B P PBP P P AP ( ).. Suppose ( B C) D, w here B ad C are m matrices ad D is ivertible. Prove that B C. 1 1 We have ( B C) D, so B C ( B C) DD D ad B C ( B C) C C. 3. Suppose A ad B are square matrices, B is ivertible, ad AB is ivertible. Prove that A is ivertible. [Hit: Let C AB, ad solve this equatio for A i terms of B ad C.] If C AB, w e have A ABB CB 1 1, so A ( CB ) ( B ) C B( AB). 4. Solve the equatio AB BC for A, assumig that A, B, ad C are square ad B is ivertible. We have A ABB BCB Asw er true or false to the follow ig. If false offer a couterexample. a. If u ad v are liearly idepedet, ad if w is i Spau, v, the u, v, ware liearly depedet. True. If w is i Spau, v it must be a liear combiatio of u ad v. b. If three vectors i 3 lie i the same plae i 3, the they are liearly depedet. 1 False. The three vectors u, v 1, w x3 1, but are liearly idepedet. are i 3 lie ad lie i the plae
2 c. If a set cotais few er vectors tha there are etries i the vectors, the the set is liearly idepedet. False. The set of the sigle vector idepedet. u has tw o etries but is ot liearly d. If a set i is liearly depedet the the set cotais more tha vectors. False. The set of the sigle vector cotais more tha vectors. u is liearly depedet but does ot e. If v 1 ad v are i idepedet. 4 ad v is ot a scalar multiple of v 1, the v 1 are liearly 1 False. With the vectors v1, v v1, v are liearly depedet sice v 1 v. 4 is ot a scalar multiple of v 1, but f. If v 1, v 3 are i idepedet. 3 ad v 3 is ot a liear combiatio of v 1, the v 1, v 3 are liearly 1 3 False. With the vectors v1, v, v 3 1, v 3 is ot a liear combiatio of v 1,, but v 1, v 3 are liearly depedet sice v 1 v v 3. g. If,,, v v v v is a liearly idepedet set of vectors i liearly idepedet. 4,the,, v v v is also 1 3 True. If o liear combiatio of the elemets of v1, v, v3, v4 combiatio of the elemets of v, v, v is zero. 1 3 is zero the o liear
3 6. Asw er true or false to the follow ig. If false offer a couterexample. a. The rage of the trasformatio x Ax is the set of all liear combiatios of the colums of A. True. The rage of the trasformatio x Ax is the set of all vectors of the form Ax ad that equals the set of al liear combiatios of the colums of A. b. Every matrix trasformatio is a liear trasformatio. True. We have A( x) Ax ad A( x y) Ax Ay so the trasformatio x Ax is liear. c. A liear trasformatio preserves the operatios of vector additio ad scalar multiplicatio. True. We have T( x) A( x) Ax T( x) ad T( x y) A( x y) Ax Ay T( x) T( y) so the operatios of vector additio ad scalar multiplicatio are preserved. m d. A liear trasformatio T : alw ays maps the origi of to the origi of m. True. Sice T() T( x) T( x) for ay vector x, a liear trasformatio m T : alw ays maps the origi of to the origi of 7. Asw er true or false to the follow ig. If false offer a couterexample. m. a. If A is a 4 3 matrix, the the trasformatio x Ax maps 3 oto 4. 1 False. Let A 1 ad b, the for ad 4 ad thus for o x is Ax = b. x 3, Ax has zeros i compoets,3, b. Every liear trasformatio from to m is a matrix trasformatio. True. Let A be a m matrix w ith colums T( e1 ),..., T( e ) (w here e 1,..., e are the colums of I ), the the matrix trasformatio x Ax is equivalet to x T( x ).
4 c. The colums of the stadard matrix for a liear trasformatio from images uder T of the colums of the idetity matrix. to m are the True. Let A be a m matrix w ith colums T( e1 ),..., T( e ) (w here e 1,..., e are the colums of I ), the the matrix trasformatio x Ax is equivalet to x T( x ). m d. A mappig T : is oe-to-oe if each vector i maps oto a uique vector i m (meaig tw o vectors i do ot map to the same vector i m ). True. A fuctio from to ot map to the same vector i m is oe-to-oe if ad oly if tw o vectors i m. do 8. Fid formulas for X, Y, ad Z i terms of I, A, B, ad C ad iverses. Assume A, B, ad C have iverses. (Hit: Compute the product o the left, ad set it equal to the right side. First, preted the blocks are simply real umbers but make sure you do ot ever divide you may multiply by iverses, how ever. Be careful about right ad left multiplicatio.) I all cases, assume the block matrix dimesios are such that the products are defied. a. X A I Y Z B C I W e have XA B I, YA ZB, X C,adY ZC I, so XA I, YA ZB, ad ZC I. We coclude that X A,Z C,ad Y ZBA C BA.
5 b. A B X Y Z I I I I We have AX B I,X I, AY B,Y I, AZ BI,ad Z I I I, so AX I, AY, ad AZ B. We coclude that 1 1 X A Z A B,Y,ad. c. I I I A I X I I B C I Y Z I I Omittig the obvious relatios, We have AI IX, BI CX IY,ad BCI IZ, so A X, B CX Y, ad C Z. We coclude that X A, Z C,ad Y B CX CA B.
6
M 340L CS Homew ork Set 6 Solutions
. Suppose P is ivertible ad M 4L CS Homew ork Set 6 Solutios A PBP. Solve for B i terms of P ad A. Sice A PBP, w e have B P PBP P P AP ( ).. Suppose ( B C) D, w here B ad C are m matrices ad D is ivertible.
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