V addition addition operation subtra operation multiplication okay so add multiplication divis okay subtraction multiplication multiplication okay okay X to the^ y + x + 1 okay so definition 2 power 5 plus 2 1 32 plus 35 okay normal addition 2 2 3 4 Mat 2 5 matrix multiplication Two element M no sir no element mul say a Comm B okay B plus d AB plus CD thenation then second ENT acus fix object okay object addition keep okay okay okay s that means 10 okay set of all set of okayi okay so numbers okay comp number a Comm such that real number second number 10 half 3 by2 combin 1 2 65 3 minus 2 so combination possible combination possible okay the vectorace vector set of vectors V okay just vectors set of reals usually reals V Vector multiplication scalar scal real number real number then Vector add r Okay add so okay say V 1 2 3 4 5 okay u v Okay small U small v u plus v v five right yes so V V plus 4us 2 2us X Plus y x to the power Y 2 3 8 then thir u v 64 8 64 then add say x x onep one VOR okay Okay so mation one okay inverse Vector okay X inverse minus x a x inverse 1 okay okay add vector vector number Clos stand scal multiplication so number okay3 scalar uh V okay then okay V vector addition okay vector vector R squ so r squ r okay simple X1 y1 multiplication multiplication suppose 15 always Z second always Z okay okay okay1 okay1 uh M2 so M5 get F okay m m 1 * = U that means 1 * U equals to U okay so left hand side left hand side equals to 1 * U okay so XY right right so 1 time U okay left hand side right so left hand side right hand side so that means M5 fails clear can M5 fail so left hand side if you multiply X1 y1 by K the result is K X1 0 so the result is 1 * 2 question okay [Music] M I think Vector is the set R square is a vector space answer is no as usual so XY Z and X Prime y Prim Z Prime is this this is regular addition so [Music] problem usually or M3 M3 M4 M3 or M4 M3 M5 okay okay definition x y z plus X Prime y Prime Z Prime equal to x + x Prime y + y Prime Z + z Prime K * x y z = to 0 0 0 this is our definition so test for M5 so left hand side left hand side is 1 * U and right hand side is just U so you particular value say u 2 3 4 okay so 1 * 2 3 4 so e is 2 three 4 so result will be 0 0 so M cor sir cor okay so this is not a vector space all but with a condition the first entry must be greater or equal Z that means with the stand that Mei question two so our Vector set V will be all pairs such that XY belongs to r with a condition X great than or equal then add x y + XE y Primal to x + x Prime y + y Prime then multiplication will be KX k y this is standard oper [Music] okay so1 okay meals toal 2A 5 this belongs to V our k k equals to uh three that belongs to our set real number or if is times s33 then -3 * 25 so result -6 15 this is not belongs to why that's why and for all U belongs to V that me okay okay okay question three question plus xe+ 1 y+ y one question x + sorry x y+ x Prime y Prime this is equals to x + x Prime + 1 y + y Prime + 1 and multiplication just regular multiplication [Music] think question two problem okay question problem question two examp left right hand left hand side right hand side 2 V 5 2 3 and 57 2 3+ 5 S so 2 + 3 2 + 5 7 plus extra one that means 8 3 + 17 11 so this will be 57 + 23 so this will be 5 2 7 + 1 8 10 11 problem problem addition a that means that means K into U + v = k k so four that means four times Vector 23 + 57 this will be K * U 23 + * V 5 okay and so six or 11 six or 11 no8 and 11 oh no no eight and 11 sorry okay so multip mation simple just regular so multiplication will be 32A 44 okay second multiplication regular so that will be 8 12 plus that will be 20 okay so okay so this is our definition for addition so one okay clear okay okay this will be so multiplication regular so multiplication so 20 28 20 28 29 or 41 yes 29 or 41 okay so so that's why I'm okay clear clear okay add X Y X + 0 + 1 y + 0 + 1 okay so that means I X + 1 y + 1 right yes sir yes sir sir okay so this is our zero Vector this is our zero Vector Z [Music] VOR say XY + x Prime y Prime equals to um so op okay okay 0 0 add then result will be X to the^ 0 + 1 y the^ 0 + 1 will be x y okay clear yes sir okay then last so last the set of all pairs of the form one y that means first slot always One V equals to all the pair such that the first is one the sign is y so only y can vary first always one that means 2 three ENT 1 three okay so first yes sir okay then y 1 1 y k no sir similarly 1 + y 1 y + y Prime this will be 1 k y so into right 0 okay okay but 0 add V so that's why a important okay M3 Pro okay vectorace Pro the question four Samra M left hand side this will our right hand side say 1 Comm x v 1 Comm X Prime Pro say two two number prime number even a u plus v right hand side a okay a times 1 X Plus 1 Comm y a * 1 x + a * 1A Sor X Prime 1 x + XE right X then 1A a x + a x Prime okay idea clear [Music] okay okay way V say R2 R2 is then this is a vector space Vector space Vector space similarly R3 R4 r x y similar R vorace R2 R3 R okay then time almost let vset a w okay Subspace V you want M yes sir okay M5 okay so A1 A2 A3 A4 a A3 M2 M3 M4 M4 M3 M2 M5 V plus right for natural number set natur number set 1 2 3 4 five 6 7 8 3 4 five1 And1 CL prop Z then M3 okay A4 A1 M1 okay V V a A1 then1 a SE number operation okay normal addition Z right okay a done A1 sor So to that means okay sub Pro easy but 5 2 but 52 1 5 Element second A1 OB first okay so 413 413 that's A1 let a b c belongs to s a prime B Prime C Prime belongs to S okay a a c belongs to S aus C = A minus b c Prime = to a c prime a + a prime b + b Prime c plus c Prime okay a a plus a prime minus B plus b Prime let a b c belongs to this cus C sorry C plus Aus okay thank you thank you okay C A minus B then second belongs to real number or field field or into ABC k a KB that's that means is a Subspace of okay I think A + B + 3 A B C real number c c zal to A + B + 3 no sir no statement okay I think clear VOR Matrix M by 2 by matx C [Music] C XY a b c X+ y + + a plus x b + y C+ Z C + z plus d + w a plus b plus C plus d X+ y + z easy all 2x two symmetric Matrix symmetric Matrix okay the symmetric Matrix symmetric Matrix It symmetric Matrix a equal and c b equal so symmetric Matrix symri Matrix symmetri Matrix a b c d such thats B B ZM zri symmet a b c a b c c a b b c next yes sir give symmetric aals so a transpose equals to A symmet aose equals to minus a a b b c such that a c belongs to R 2 [Music] by a cus ausus C 33 Okay 3 by3 3 symmetric symmetric symetric 2 by a b b c 33 a b c d e f g h reverse okay okay okay XY [Music] first entry 2 into second entry first entry 2 into second entry plus 3 into third entry equals to 5 all blah blah blah XY Z with X plus 2 y 3 equals 0 0al XY 0 satisfy both a 2 3 x + a y + b z + C then satisfy x + a plus y + b 2 into y + B + 2 y + 3 z+ a + 2 b + clear idea yes sir okay all positive reals positive reals if always real number s M4 M4 okay M4 a plus b u a b left hand right M for a plus b u a u plus b okay a real number number real number just just number okay so a plus b a clear hello sir b a left hand right hand side okay a A plus b real number real number real number r+ set of scal okay