<prompt> Mabadiliko T, kuchukua vekta kwa vekta, ina sifa zifuatazo:i Ta mathbfv b mathbfw a Tmathbfv b Tmathbfw kwa vekta zote mathbfv na mathbfw, na kwa alama zote a na b.ii Tmathbfv nyakati mathbfw Tmathbfv mara Tmathbfw kwa vekta zote mathbfv na mathbfw.iii T anzapmatrix 6 6 3 mwishopmatrix anzapmatrix 4 -1 8 mwishopmatrix.iv T anzapmatrix -6 3 6 mwishopmatrix anzapmatrix 4 8 -1 mwishopmatrix.Tafuta T anzapmatrix 3 9 12 mwishopmatrix.  <response>: Kutoka ii, iii, na iv,T kushoto anzapmmatrix 6 6 3 mwishopmatrix nyakati anzapmatrix -6 3 6 mwishopmatrix kulia anza pmatrix 4 -1 8 mwishopmatrix nyakati anzapmatrix 4 8 -1 mwishopmatrix.Hii inapunguza hadiT anzapmatrix 27 -54 54 mwishopmatrix anzapmatrix -63 36 36 mwishopmatrix.Hasa, kutoka i , T a mathbfv a Tmathbfv. Kwa hivyo, tunaweza kugawanya vekta zote mbili kwa 9, ili kupataT anzapmatrix 3 -6 6 mwishopmatrix anzapmatrix -7 4 4 endpmatrix.Sasa, tunaweza kujaribu kueleza beginpmatrix 3 9 12 endpmatrix kama mchanganyiko wa mstari ufuatao:anzapmatrix 3 9 12 mwishopmatrix a anzapmatrix 6 6 3 mwishopmatrix b anzapmatrix -6 3 6 mwishopmatrix c anzapmatrix 3 -6 6 mwishopmatrix anzapmatrix 6a - 6b 3c 6a 3b - 6c 3a 6b 6c mwishopmatrix.Kutatua 6a - 6b 3c 3, 6a 3b - 6c 9, na 3a 6b 6c 12, tunapata a frac 43, b 1, na c frac13. Hivyo,anzapmatrix 3 9 12 mwishopmatrix frac43 anzapmatrix 6 6 3 mwishopmatrix anzapmatrix -6 3 6 mwishopmatrix frac13 anzapmatrix 3 -6 6 endpmatrix.Kisha kwa i,T anzapmatrix 3 9 12 mwishopmatrix frac43 anzapmatrix 4 -1 8 mwishopmatrix anza pmatrix 4 8 -1 mwishopmatrix frac13 anzapmatrix -7 4 4 endpmatrix anzapmatrix 7 8 11 endpmatrix.Kwa kazi zaidi, inaweza kuonyeshwa kuwaT anzapmatrix x y z endpmatrix newcommandarraystretch1.5 beginpmatrix -frac727 frac26 27 -frac227 -frac1427 -frac227 frac2327 frac22 27 frac727 frac1427 endpmatrix renewcommandarraystretch1 anzapmatrix x y z mwisho pmatrix.Kwa kazi zaidi, inaweza kuonyeshwa kuwa T ni mzunguko katika nafasi. 