![Aus 170) to varify the operation (x,y) = x,, - x, I, e – X2Z, + 3x222 =(2, X2) and I = (4 ,42) is an innerproduct n cchere R?](//img.homeworklib.com/questions/a6479560-22b0-11eb-8c20-df81db7c8819.png?x-oss-process=image/resize,w_560)
![-c (2,4, -*%, - X,%, + 3% %) = c < x,y) c) <**> x, y, - X, Y, – X,Z, + 3x₂92 (as 2, 2, x .7. ER) X,Y, - x, Z - X, Y, + 3 X2 %](//img.homeworklib.com/questions/a7f48fc0-22b0-11eb-9932-a3f7620ddeca.png?x-oss-process=image/resize,w_560)
![then x = 0 and X₂=0 and X3²0 Ans 1.72) set x = (x, % 2, ), I = (3,3,7) & 2=(2,, Z2, Z3) ER & CER x,J,+ 2x2 + x3 73 i) <x+3, y](//img.homeworklib.com/questions/a9c81e10-22b0-11eb-a09c-b95f635d7765.png?x-oss-process=image/resize,w_560)
![x = 0 0 < x,x) 20 a) sotesfies oll the property so it is inner product b) < x,x) = x;%; + x,%; + x, Now <cx, y) = ((x,3%,](//img.homeworklib.com/questions/ab4e4540-22b0-11eb-aa28-89cc37f33a88.png?x-oss-process=image/resize,w_560)
Aus 170) to varify the operation (x,y) = x,, - x, I, e – X2Z, + 3x222 =(2, X2) and I = (4 ,42) is an innerproduct n cchere R? on (x,x), y = (33) =(2, 2) ER² x = Now det and CER a) < x + 2, 4) <(x, +2, & +22), (3, 1)) (x + 2) Y, (M,+z)%, - (X, +22)%, + 9(x₂ + 2)(x) x, y, + E, I, - x, Z – 2,4 – HZ, - 2,4, + 3₂ + 3 2₂ 22 =xY, –a, Z - XqZ, + 3 X₂ Z 2 + 2,4 – 2,7 - 2₂9, +3 22 22 <x, 2) + {2,4) <cCX, X), CY,,%) b) <cr,y) < (Cx,,CX), (3, 2)) cx, y, - CX, Y, - CR₂4, + 3CX 222
-c (2,4, -*%, - X,%, + 3% %) = c < x,y) c) <**> x, y, - X, Y, – X,Z, + 3x₂92 (as 2, 2, x .7. ER) X,Y, - x, Z - X, Y, + 3 X2 % 9, X, Y, Z, X ₂ + 3 Yk2 y, x, - Y, H₂ - YX, + 372 <3, x) d) {x, x) = x,x - X,H2 - X,X, +3x, X2 - *. X,'+33,2 x 7o x > 0 xz?> 0 2₂ o if if x² o <x,x) > 0 از x=0 then 2,80 & X2=0 <x,x) = 0²+3.0² as all the properties o, b, c, d ore b, c, d ore sotesfied by <x,y) so the defined operation is on vaner product.
then x = 0 and X₂=0 and X3²0 Ans 1.72) set x = (x, % 2, ), I = (3,3,7) & 2=(2,, Z2, Z3) ER & CER x,J,+ 2x2 + x3 73 i) <x+3, y) = (x,+z) 4, +2C x2 +2.)%, + (x3 +2.) 73 x, 7, +2,5, + 2X₂Z + 2 ₂ 2 ₂ + X37, + 27% = x, y, +242 + Z₂ Z3 +2,7, + 22, 2, + ZzZz +19%) <x,y)+(2,4) ii) <cx,y) = (2x)), +2 (622) J. + C CX2723 (2,2, +212 % + =c<x, y) tii) {x,y) = 2,9, +2 x 2 + x 33 x, G, +2₂7 + X₂ 73 = 4x + 2% X₂ + 2X₂ <y, x) x x + 2 X₂, X2 + XX, x,?+ 2x' + x if x 7o then x,70 or X, 70 or X₂ #0 :. x > 0 on x² > 0 o xo x > 0 iv) < x,x) < x,x) = so if x=0
x = 0 0 < x,x) 20 a) sotesfies oll the property so it is inner product b) < x,x) = x;"%;' + x,'%;' + x, Now <cx, y) = ((x,3%, '+ ((x,) %, ' + ((*)&,? 2 » c²(x, y," + x, y + x, y) ² < x, y) & cx,y) is not a inner product. the defined <x,y) c) (x,y) = x,y, - X₂Z + X₂Yz x=(1,1,0) che home <x,x) = 1.1-1.1 -0.0 but xo 1-10 ů a the defined (x,y) is not a inner product. d) (x,y) = 8, 9, + X₂Z2 for x = (0,0,1) awe have <x,x) = 0.0 + 0.0 = 0 bet xto o the defined {x,y) is not a innes product.