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Question
Good day Prof,

I am stuck on question on Linear Transformations.

Can you please shed some light. I will really appreciate this.

Consider T: P2 to M22 given by T(a+bx+cx^2) = [a11=2a       a12=b        a21=b     a22=2c] for all a,b,c element of R.

3.1 Show that T Is a linear transform

3.2 Find the matrix representation for T relative to the standard basis in P2 and in M522 with the usual ordering.

3.3 Is T invertible?

3.4 Show that the range of T is the subspace M22(with dash above M) of M22 consisting of symmetric matrices.

3.5 Let T (with dash above T): P2 to M22(with dash above M) be defined by T (with dash above T)(p(x)) for all p(x) E P2. Find the matrix representation for T (with dash above T) relative to the standard basis with the usual ordering in P2 and the basis
{ [a11=1   a12=0   a21=0   a22=0], [a11=0  a12=0   a21=0 a22=1], [a11=0   a12=1   a21=1   a22=0]}
For the symmetric matrices, ordered left to right.

Thank you

Answer
Just to make sure we understand the notation and nomenclature, P2 is the vector space of polynomials of degree 2 or less and has a standard basis

(1, x, x^2).

M22 is the vector space of 2x2 matrices and has a standard basis

m11  = ( 1  0  )
         ( 0  0  )

m12  = ( 0  1  )
         ( 0  0  )

m21  = ( 0  0  )
         ( 1  0  )

m22  = ( 0  0  )
         ( 0  1  ),

since any linearly independent and that any 2x2 matrix can be written

( a  b )
( c  d )  = a・m11 + b・m12 + c・m21 + d・m22.

3.1 To show that T is a linear transform just show that, if P2,1 = a1,a2x,a3x^2 and P2,2 b1,b2x,b3x^2, then T(P2,1+P2,2) = T(P2,1)+T(P2,2).

3.2  The linear transformation T, defined as the 2x2 matrix with a,b and c as given in your question takes P2 polynomials and transform them into 2x2 matrices. It can be decomposed into the sum of the scalars a, b and c and 2x2 matrices M1, M2, M3 (defined in the following)

a・(1 0) + b・(0   1/2) + c・(0 0)
  ( 0 0)        (1/2 0  )        (0 1)   =  aM1 + bM2 + cM3.

which can be written

(A)   a・m11 + b・(m12 + m21) + c・m22.

The matrices M1, M2 and M3 are the images in M22 of the basis vectors 1,x,x^2 inP2.

Note that the mij are really 2x2 matrices but also correspond to vector components so that (A) can be written as a a big matrix representation of T for P2 -> M22

a(1,0,0,0), b(0,1/2,1/2,0),c(0,0,0,1)・(m11,m12,m21,m22)^t.

3.3 I think T is invertible since det(T) ≠ 0.

3.4 Range of T is spanned by the image vectors so that

V = a1M1 + a2M2 + a3M3 represents a vector in the range on T (for arbitrary aj). These are symmetric vectors since, if you plug in the values for the matrices, you can show

V = V^t.

3.5 As before, the T matrix can be decomposed into the basis "vectors" (really 2x2 matrices) M'1, M'2 and M'3 as

M'1 = a・m11, M'2 = b/2・(m12+m21) and M'3 = c・m22

Hope this helps. Its a lot to swallow.

Randy  

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