Linear Algebra Answers

Define T : R
3 → R
3 by
T(x, y, x) = (−x, x−y,3x+2y+z).
Check whether T satisfies the polynomial (x−1)(x+1)
2
. Find the minimal
polynomial of T.

Let P superscript (e) ={p(x)∈R[x]|p(x) = p(−x)} P superscript(o) ={p(x)∈R[x]|p(x) =−p(−x)} a) Check that P superscript (e) and P superscript (o) are subspace of R[x]. b) Show that P superscript (e) =(∑ i a subscript i x superscript i ∈R[x]



 a subscript = 0 if i is odd.) P superscript(o) =(∑ i a subscript i x superscript i ∈R[x]



 a subscript = 0 if i is even.) Deduce that P superscript (o)∩P superscript (e) ={0}. ( c) Check p(x)+p(−x)∈P superscript(e) for every p(x)∈R(x). Check that the map ψ: R[x]→P superscript(e) given byψ(p(x)) = p(x)+p(−x)/ 2 is a linear map. Further, check that ψ superscript 2 =ψ. Determine the kernel of ψ

Check whether the matrices A and B are diagonalisable. Diagonalise those matrices which are diagonalisable. i) A =  1 0 0
1 2 −3
1 1 −2  
ii) B =  −2 −4 −1
3 5 1
1 1 2 .b) Find inverse of the matrix B in part a) of the question by using Cayley-Hamiltion theorem. c) Find the inverse of the matrix A in part a) of the question by finding its adjoint.

The following is the reduced row echelon form of the augmented matrix for a system of linear equations.

1 0 -3 0 0 | 2
0 1 -1 0 2 | 6
0 0 0 1 1 | 3
0 0 0 0 0 | 0

a) How many variables were in the original system of equations, and what tells you this?

b) How many equations were in the original system of equations, and what tells you this?

c) State the solution (unique or general as appropriate), if it exists, for the system using x1,x2,x3,… for variables. If no solution exists, explain why no solution exists.

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)=x1^2-2x1x2+4x2x3+x2^2+x3^2 , find the representation of Q in terms of (y1,y2,y3).

Consider the linear operator T:C^3 be defined by
T(z1,z2,z3)= (z1+iz2, iz1-2z2+iz3,-iz2+z3).
Compute T^* and check whether T is self-adjoint.
Check whether T is unitary.

Find the orthogonal canonical reduction of the quadratic form -x^2+y^2+z^2-4xy-4xz.
Also, find it's principal axes.

Let T:R^3 be defined by
T(x1,x2,x3)=( x1+x3,x3,x2-x3).
Is T invertible? If yes,find a rule for T^-1 like the one which defines T. If T is not invertible, check whether T satisfies Cayley-Hamiltion theorem.

Consider the basis e1=(1,1,-1), e2=(-1,1,1) and e3=(1,-1,1) of R^3 over R. Find the dual basis of {e1,e2,e3}.

Let T: P2 be defined by
T(a+bx+cx^2)= b+2cx+(a-b)*x^2.
Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1={x^2, x^2+x, x^2+x+1} and B2= {1,x,x^2} . Find the kernel of T.