Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1={(1,0,0),(0,1,0),(0,0,1)},B2={(1,0,0),(0,1,2),(0,2,1)}.If Q(x)=x1²+2x1x2+2x2x3+x2²+x3²,find the representation if Q in terms of (y1,y2,y3).?

2 Answers
Ratnaker Mehta
Jan 27, 2018

#Q(y_1,y_2,y_3)=y_1^2+9y_2^2+9y_3^2+2y_1y_2+18y_2y_3+4y_3y_1#.

Explanation:

#B_1={i=(1,0,0),j=(0,1,0),k=(0,0,1)}# is a Basis for

#x=(x_1,x_2,x_3)=x_1i+x_2j+x_3k#.

Also, #B_2={baru=(1,0,0),barv=(0,1,2),barw=(0,2,1)}# serves as

the Basis for #y=(y_1,y_2,y_3)#, so that,

#y=(y_1,y_2,y_3)=y_1baru+y_2barv+y_3barw#.

But,

#baru=(1,0,0)=1i+0j+0k#,

#barv=(0,1,2)=0i+1j+2k, and #

#barw=(0,2,1)=0i+2j+1k#.

#:. y=(y_1,y_2,y_3)=y_1baru+y_2barv+y_3barw#,

#=y_1(1i+0j+0k)+y_2(0i+1j+2k)+y_3(0i+2j+1k)#,

#=(y_1i)+(y_2j+2y_2k)+(2y_3j+y_3k)#,

#=y_1i+(y_2+2y_3)j+(2y_2+y_3)k#.

Knowing that, #Q(x)=Q(x_1,x_2,x_3)=Q(x_1i+x_2j+x_3k)#,

#=x_1^2+2x_1x_2+2x_2x_3+x_2^2+x_3^2#,

#=x_1^2+x_2^2+x_3^2+2x_1x_2+2x_2x_3color(red)(+2x_3x_1-2x_3x_1)#

#:. Q(x_1i+x_2j+x_3k)=(x_1+x_2+x_3)^2-2x_3x_1#.

Accordingly, #Q(y)=Q(y_1i+(y_2+2y_3)j+(2y_2+y_3)k)#,

#={y_1+(y_2+2y_3)+(2y_2+y_3)}^2-2(2y_2+y_3)y_1#,

#=(y_1+3y_2+3y_3)^2-2(2y_2+y_3)y_1#,

#=(y_1^2+9y_2^2+9y_3^2+6y_1y_2+18y_2y_3+6y_3y_1)-4y_1y_2-2y_3y_1#,

#rArr Q(y_1,y_2,y_3)=y_1^2+9y_2^2+9y_3^2+2y_1y_2+18y_2y_3+4y_3y_1#,

is the desired expression!

Enjoy Maths., & Spread the Joy!

Cesareo R.
Jan 27, 2018

See below.

Explanation:

Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases

#B_1=((1,0,0),(0,1,0),(0,0,1))#

#B_2=((1,0,0),(0,1,2),(0,2,1))#

#Q(x)=x_1²+2x_1x_2+2x_2x_3+x_2²+x_3²#

Here #x = B_2 y#

and

#Q(x) = x^top((1,1,0),(1,1,1),(0,1,1))x = x^top M x#

then

#x = B_2 y# and

#Q(y) = y^top B_2^top M B_2 y# or

#Q(y) = y_1^2+9y_2^2+y_3^2 +2 y_1y_2+18 y_2y_3+4 y_1y_3#

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