Linear Algebra Answers

Let T : P2 → P1 be defined by
T(a+bx+cx2) = b+c+(a−c)x.
Check that T is a linear transformation. Find the matrix of the transformation with
respect to the ordered bases B1 = {x2,x2+x,x2+x+ 1} and B2 = {1,x}. Find the
kernel of T .

Reduce the conic x2 − 6xy+y2 − 4 = 0 to standard form. Hence the given conic.

Find the orthogonal canonical reduction of the quadratic form
−x2+y2+z2+ 2xy− 2xz+ 2yz. Also, find its principal axes

Find the orthogonal canonical reduction of the quadratic form
−x2+y2+z2+ 2xy− 2xz+ 2yz. Also, find its principal axes

Find the orthogonal canonical reduction of the quadratic form
−x2+y2+z2+ 2xy− 2xz+ 2yz. Also, find its principal axes

Solve and Check
For (1)
2a+b-3c-d= -2
a-b-c+3d= 0
3a+2b+c+5d= 6
a-c-d= 2

and (2)
r+s+2t-u= -3
2r+3s+3t+u= 2
4r+2s-t+u= 5
s+2t+2u=7

Q. Show that the transformation between the coordinates X1, X2, X3 and X1’ , X2’ , X3’ defined by
X1’=1/3(2x1+2x2-x3)
X2’=1/3(2x1-x2+2x3)
X3’=1/3(-x1+2x2+2x3)
Is orthogonal and left-handed(improper)

Q. A vector A in OX1X2X3 has components (2, 1, -2). Find its components in OX1’X2’X3’. The transformation between the coordinates X1, X2, X3 and X1’ , X2’ , X3’ is defined by
X1’=1/3 (2x1+2x2-x3)
X2’=1/3 (2x1-x2+2x3)
X3’=1/3(-x1+2x2+2x3)

Q. Show that the transformation between the coordinates X1, X2, X3 and X1’ , X2’ , X3’ defined by
X1’=1/3 (2x1+2x2-x3)
X2’=1/3 (2x1-x2+2x3)
X3’=1/3(-x1+2x2+2x3)
Is orthogonal and left-handed(improper)

B = [ 0 a 0 0]
b 0 0 0
0 0 c 0
0 0 0 d
Let Bn the ( n x n) submatrix in the TOP left hand corner of B. Define B1, B2, B3 and B4. Compute determinate of B1, B2 , B3 and B4. Find conditions of a, b, c, d such that 4 determinants cannot be negative .