Linear transformation. TEST today help.

[trap101]

Not too big of a question, it's more so just handling a certain presentation.

T: M_2x2-->M_2x2 defined by: T(A) = (A + A^T)/2. So my question is, how do I handle the fraction considering it will be a matrix?

 

[Dick]

Not too big of a question, it's more so just handling a certain presentation.

T: M_2x2-->M_2x2 defined by: T(A) = (A + A^T)/2. So my question is, how do I handle the fraction considering it will be a matrix?

The '2' isn't a matrix. It's a scalar. Write it as (1/2)*(A+A^(T)). Nothing complicated here.

 

[trap101]

Thanks. I have another quick question. I'm finding a transformation matrix [T]_ββ .

The form is 2x2...I've gotten through everything including obtaining the values for the coeffcient vectors. I have two vectors:

\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & 2 \ end{bmatrix}

how do I convert these into the [T]_ββ...since I know that has to be 2x2?

 

[HallsofIvy]

You seem to be very confused- or a little loose with your terminology. What you show is not "two vectors", it is a matrix. And before you can write a linear transformation, you will have to define the linear transformation. Then, a linear transformation, from vector space U to vector space V, can be written as a matrix given specific bases for both U and V. Of course, if U and V are both R², it would be standard to take <1, 0> and <0, 1> but since you talk about $[T]_{\beta\beta}$, I take it that you are given some basis and I suspect that is what <1, 0> and <0, 2> are- you are asked to find the matrix form of linear transformation T in that basis. But that still leaves the most crucial question- what is T? How is T defined?

Or are you using the standard basis, <1, 0> and <0, 1>, and have determined that T(<1, 0>)= 1<1, 0>+ 0<0, 1> and T(<0, 1>)= 0<1, 0>+ 2<0, 1>? In that case, the matrix you give is the matrix form you are looking for.

 

[trap101]

Let T: C2 ! C2 be the linear transformation whose matrix with respect to the stan-
dard basis of C2 is 

(2x2) Matrix with complex numbers

Find a basis  for C2 consisting of eigenvectors of T and [T]_ββ.

That's the question.


So I've obtained eigenvectors and applied the eigenvectors (which resulted in 2x2 matrices) to the transformation matrix I was given. This produced a new 2x2 matrix. Now I wanted to write out this new 2x2 matrix as a linear combination of my standard basis and take those co-efficients to construct a new transformation matrix.

 

[trap101]

Nevermind, I figured out my mistake. It might help if I did actually get the eigenvectors, instead of the matrix A-I(lambda)