Worsheet # th Nme:. Sole the folloing sstems of liner equtions. he proe tht the solutions forms suspe of ) ). Find the neessr nd suffiient onditions of ll onstnts for the eistene of solution to the sstem:. Gien the mtri Wht must e true out nd if n e redued to the identit mtri.. Gien tht ) Sho tht is not inertile. ) Sole the homogenous sstem. Sole the sstem using the inerse of the oeffiient mtri.. Find the determinnt of the mtri Is inertile?. Let ) Find det ) For ht lue of ill he n inerse? ) For those suh tht eists find det. Use determinnts to find n eqution of the irle through points nd 9. Find ll lues of for hih the mtri is nonsingulr then find
g j. ssume tht d e f is suh tht det ) Is nonsingulr? If so ht is det ) Elute det( ) h det ) Let d e f g h j g h j. Find det().. roe the folloing: ) Let e n n n mtri nd n mtri suppose tht Y nd Z re to different n mtries for hih Y= nd Z=. i) Use this informtion to sho tht the sstem X= hs nontriil solution. ii) Conlude from prt (i) hether or not is inertile. ) Let e n n smmetri mtri nd e nother n n mtri then is lso smmetri. ) Eer inertile smmetri mtri hs smmetri inerse.. ) Let... e olletion of etors in etor spe V. n roe tht Spn is suspe of V. ) Let d e olletion of polnomils in n n. roe is suspe of n ) Let roe or disproe is suspe of n d) Let det. roe or disproe is suspe of e) d d. roe or disproe is suspe of f) Let e the etor spe of ll polnomils in ith rel oeffiients. Let S f ll the eponents of in f re een. Is S suspe of? Eplin. n. g) f f h) i) C. Whih of the folloing etors form sis for ) nd ) nd ) nd d) nd ' ' '?
. Find minimum set of tht spn() genertes the sme etor spe: ) 9 ) ). roe or disproe the folloing suset is suspe of. Find sis if suspe. ) ) Let ) d) Let e) f) Let g) d h). ) Let. Desrie the spe Spn. Does the etor lie in this spe? ) roe tht spns. ) For ht lues of if n is in the spn of d) In onsider N. Determine hether N is in the spn of e) If u is linerl independent suset of etor spe V. Sho tht u u is lso linerl independent set.
. ) roe or disproe: is sis for ) Consider the etor spe. Sho tht the folloing set of etors form sis for. hen find the oordinte etor (omponents) of the etors:. nd d ) Find the dimension of the suspe of nd spnned d d) roe tht the set S of S. is suspe of. hen find sis e) Find sis nd dimension of the suspe S re rel numers f) Find sis if the null spe of g) Find sis of S tr (tr()=sum of ll entries long the min digonl) h) Find sis of S.. ) Sho tht u u is n inner produt spe under the folloing definition of inner produt. u u ) Let nd u u u u nd define n opertion u u u u. Determine if this opertion is n inner produt on nd if it is not stte hih ondition fil. ) he tre of squre mtri is denoted tr nd is defined to e the sum of the entries on the min digonl of. Let nd e elements of n inner produt on n 9. Find sis for the rospe() nd olspe() here n. Sho tht tr defines
. Find sis of ernel nd rnge of the folloing liner trnsformtion. ) V V : defined for V. ) : defined ) is defined d) n : defined tr e) ) ( : i) Determine hether p is in the rnge of. ii) Determine hether p is in the ernel of.. Determine mtri of representtion of the folloing liner trnsformtion ith respeted to indited sis nd then determine sis of Ker() nd ng() ) : ith stndrd sis. ) : ith nd ) : ith nd d) d df f : ith stndrd ses.. Find the Eigenlues nd ssoited Eigenetors for the folloing mtries: ) ) ). Let ) Find ll eigenlues of. ) For eh eigenlue find s mn linerl independent eigenetors s possile. ) If is digonlile find nonsingulr mtri nd digonl mtri D suh tht D. Let ) Find digonl form of the mtri. ) Find n inertile mtri suh tht is digonl.