The discussion revolves around the matrix representation of a linear transformation between two vector spaces, specifically addressing the transformation from a 3-dimensional space to a 2-dimensional space. Participants are exploring the implications of the dimensions of the matrix and the bases involved.
Some participants have provided insights into the requirements for the matrix dimensions and the nature of the transformation. There is ongoing exploration of the transformation's definition and its implications for the matrix entries, with no explicit consensus reached yet.
Participants are working with specific bases for the vector spaces involved, and there is mention of a polynomial representation that may influence the understanding of the transformation. The discussion reflects a need for clarity on the definitions and operations related to the transformation.
GwtBc said:Ok I can see the logic that led you through this attempt but it's not quite correct. First off you have a 3x3 matrix for a transformation that goes from a 3-D space to a 2-D one. This is already troublesome. You want your number of columns to be equal to ##\dim{U}## and the number of rows to be equal to the dimension of your target space i.e. ##\dim{V}##. So in this case you want a 2x3 matrix.
As for the entries of the matrix, each column is directed by letting ##T## act on a basis vector from ##U##. For example, if we have the vector ##
\begin{pmatrix} 1\\ 0\\ 0
\end{pmatrix}## which represents a constant polynomial equal to ##1##, then ##T(
\begin{pmatrix} 1\\ 0\\ 0
\end{pmatrix})## gives us ##
\begin{pmatrix}
1\\
0
\end{pmatrix}## as represented by the basis ##F## in the space ##V##.
That's just how ##T## is defined. It's a transformation that takes a vector from a 3-D space to a 2-D space. In terms of the actual calculation, you let ##T## operate on a basis vectors of E (which is a basis for ##U##). So the first one is ##\begin{pmatrix} 1\\ 0\\0 \end{pmatrix}## which represents the polynomial ##f(t) = 1##. Then ##f(3) = 1## and ##f'(3)=0##, which if represented by the basis vectors of ##F## (which is a basis for ##V##) is ##\begin{pmatrix} 1\\ 0 \end{pmatrix}##Robb said:Ok, I guess I'm unsure of
##T(
\begin{pmatrix} 1\\ 0\\ 0
\end{pmatrix})## gives us ##
\begin{pmatrix}
1\\
0
How does this go from a 3x1 to a 2x1 vector. I'm not seeing the math behind it.