Linear Algebra Answers

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

Which of the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.
i) The relation ∼ defined by R by x ∼ y if x ≥ y is an equivalence relation.
ii) If S1 and S2 are finite non-empty subsets of a vector space V such that [S1] = [S2],
then S1 and S2 have the same number of elements.
iii) For any square matrix A, ρ(A) = det(A)
iv) The determinant of any unitary matrix is 1.
v) If the characteristic polynomials of two matrices are equal, their minimal
polynomials are also equal.
vi) If the determinant of a matrix is 0, the matrix is not diagonalisable.
vii) Any set of mutually orthogonal vectors is linearly independent.
viii) Any two real quadratic forms of the same rank are equivalent over R.
ix) There is no system of linear equations over R that has exactly two solutions.
x) If a square matrix A satisfies the equation A2 = A, then 0 and 1 are the eigenvalues of
A.

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 .

given three vectors x1= transpose [ 2 4 8], x2= transpose [ 1 -1 1], x3= transpose [1 1 4]. can you write x1 as linear sum of x2 and x3?