Matrix of a linear transformation

[stunner5000pt]

This is how the question appears in my textbook
Find the matrix of T corresponding to the bases B and D and use it to compute C_{D}[T(v)] and hence T(v)

T; P2 - > R2
T(a + bx + cx^2) = (a+b,c)
B={1,x,x^2}
D={(1,-1),(1,1)}
v = a + bx + cx^2

ok i cna find Cd no problem it is
C_{D}[T(v)] = \frac{1}{2} \left(\begin{array}{ccc} 1&1&-1 \\ 1&1&1 \end{array}\right)

now am i supposed to solve for X where
C_{D}[T(v)] X = T(v)

but textbook doesn't do that...
it does C_{D}[T(v)] \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = X

It doesn't make sense... doesn't the question ask to compute Cd[T(v)] and T(v) from it??

Next question
Verify this theorem for the given transformation and uses the standard basis in Rn
Theorem: M_{EB} (ST) = M_{ED} (S) \bullet M_{DB} (T)

T;R3 -> R4
S: R4->R2
T(a,b,c) = (a+b,b+c,c+a,b-a)
S(a,b,c,d) = (a+b,c-d)

M_{EB} (S) = \left[C_{D}\left(\begin{array}{c} 1 \\ 0 \end{array}\right) \ C_{D}\left(\begin{array}{c} 1 \\ 0 \end{array}\right) \ C_{D}\left(\begin{array}{c} 0 \\ 1 \end{array}\right) \<br /> C_{D}\left(\begin{array}{c} 0 \\ -1 \end{array}\right)\right]
M_{EB} (S) = \left[\begin{array}{cc} 1&amp;0 \\ 1&amp;0 \\ 0&amp;1 \\ 0&amp;-1 \end{array}\right]

M_{DB} (T) = \left[C_{D}\left(\begin{array}{c} 1 \\ 0 \\1 \\ -1 \end{array}\right) \ C_{D}\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ 1 \end{array}\right) \ C_{D}\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \end{array}\right)\right]

M_{DB} (T) = \left[\begin{array}{cccc} 1&amp;0&amp;1&amp;-1 \\ 1&amp;1&amp;0&amp;1 \\ 0&amp;1&amp;1&amp;0 \end{array}\right]

the dimensions of the matrices are not correct though..

is there something wrong in the way i am forming the matrices?

 

[Hurkyl]

What does C_D[T(v)] mean?

First off, did you notice that T(v) is a vector in R²?


I think it might help you a great deal to write down explicitly exactly what type everything is in your equations. E.G. you might write down:

v : element of P2
T : transformation from P2 --> R²
T(v) : element of R²
C_D[T(v)] : 2x1 matrix.


I'm going to assume C_D[w] means the coordinate representation of the vector w, with respect to the basis D.

So you didn't compute C_D[T(v)] at all -- instead, you computed M_{B,D}[T] which is the coordinate representation of the linear transformation T, with respect to the bases B and D!

You really ought to go reread the section on coordinates and bases. The whole point of coordinates is that they simply transform the equation into matrices. In other words, the equation:

T(v) = w

is true if and only if the equation

M_{B,D}[T] \cdot C_B[v] = C_D[w]

is true.

 

[Hurkyl]

is there something wrong in the way i am forming the matrices?

Well, you're certainly not internally consistent:

M_{DB} (T) = \left[C_{D}\left(\begin{array}{c} 1 \\ 0 \\1 \\ -1 \end{array}\right) \ C_{D}\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ 1 \end{array}\right) \ C_{D}\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \end{array}\right)\right]

First, you wrote down the matrix with three columns, each of which is supposed to be the coordinate representation of a vector in R^4.

M_{DB} (T) = \left[\begin{array}{cccc} 1&amp;0&amp;1&amp;-1 \\ 1&amp;1&amp;0&amp;1 \\ 0&amp;1&amp;1&amp;0 \end{array}\right]

And then you wrote down a matrix whose rows are the transpose of those coordinate representations.


Incidentally, based on how the problem's stated, E is the basis on R^4, and D is the basis on R³, so you meant M_{ED}(S) and not M_{EB}(S).

 

[stunner5000pt]

Well, you're certainly not internally consistent:



First, you wrote down the matrix with three columns, each of which is supposed to be the coordinate representation of a vector in R^4.



And then you wrote down a matrix whose rows are the transpose of those coordinate representations.


Incidentally, based on how the problem's stated, E is the basis on R^4, and D is the basis on R³, so you meant M_{ED}(S) and not M_{EB}(S).

well then what is the different between M_{ED}(s) and M_{EB}(S) is it even possible to form the former?