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) = 2x^2+2x1x2-2x2x3+x2^2+x3^2, find the representation of Q in terms of ( y1, y2 ,y3 ).
To find the representation of the quadratic form Q(x) in terms of (y1, y2, y3), we need to find a change of basis matrix that transforms the coordinates from B1 to B2.
First, we need to find the change of basis matrix [T] that transforms the basis vectors of B1 to B2. This can be done by expressing the basis vectors of B2 in terms of the basis vectors of B1.
We have B1 = { (1,0,0),(0,1,0),(0,0,1)} and B2 = { (1,0,0),(0,1,2),(0,2,1) }.
To transform the basis vector (1,0,0) of B1 to B2, we can express it as a linear combination of the basis vectors of B2:
(1,0,0) = a(1,0,0) + b(0,1,2) + c(0,2,1)
By comparing the corresponding components, we get the following equations:
1 = a
0 = b + 2c
0 = 2b + c
Solving these equations, we find a = 1, b = -2, and c = 1.
Similarly, we can find the coefficients for the other basis vectors of B1:
(0,1,0) = a'(1,0,0) + b'(0,1,2) + c'(0,2,1)
=> 0 = a'
1 = b' + 2c'
0 = 2b' + c'
By solving these equations, we find a' = 0, b' = 1, and c' = -2.
(0,0,1) = a''(1,0,0) + b''(0,1,2) + c''(0,2,1)
=> 0 = a''
0 = b'' + 2c''
1 = 2b'' + c''
By solving these equations, we find a'' = 0, b'' = 1, and c'' = 1.
Hence, the change of basis matrix [T] is given by:
[T] = [1 0 0]
[-2 1 1]
[1 1 1]
Now, we can rewrite Q(x) in terms of the coordinates (y1, y2, y3) using the change of basis matrix [T]:
Q(x) = 2x^2 + 2x1x2 - 2x2x3 + x2^2 + x3^2
Let's rewrite this equation in matrix form:
Q(x) = [x1, x2, x3] * [2 1 0] * [x1]
[1 1 0] [x2]
[0 2 1] [x3]
Since [T] transforms from B1 to B2, we need to transform the coordinates (x1, x2, x3) to the coordinates (y1, y2, y3) using the inverse of [T].
[T]^-1 = [1 0 0]
[3 -1 1]
[-2 1 0]
Now we can rewrite Q(x) in terms of the coordinates (y1, y2, y3):
Q(x) = [x1, x2, x3] * [2 1 0] * [x1]
[1 1 0] [x2]
[0 2 1] [x3]
= [x1, x2, x3] * [2 1 0] * [T^-1] * [y1]
[1 1 0] [y2]
[0 2 1] [y3]
Simplifying this expression, we get:
Q(x) = [x1, x2, x3] * [2 1 0] * [1 0 0] * [y1]
[3 -1 1] [y2]
[-2 1 0] [y3]
Multiplying the matrices, we get:
Q(x) = [x1, x2, x3] * [2 1 0] * [y1]
[y2 + 3y1 - 2y3]
[y1 + y2 + y3]
Expanding the product, we have:
Q(x) = 2x1y1 + (x1y2 + 3x1y1 - 2x1y3) + (x2y1 + x2y2 + x2y3) + (0) + (0) + (0)
Simplifying further, we get:
Q(x) = 2x1y1 + 3x1y1 - 2x1y3 + x1y2 + x2y1 + x2y2 + x2y3
Therefore, the representation of Q in terms of (y1, y2, y3) is:
Q(y) = 2y1^2 + (3y1 - 2y3)y1 + y1y2 + y2^2 + y2y3