LT.M10
Define two linear transformations : \[ T:C^4 \rightarrow C^3 \] \[ S:C^3 \rightarrow C^2 \]
By :
\[ S \left(\begin{array}{c} x_1\\ x_2 \\ x_3 \end{array}\right) = \left(\begin{array}{ccc} x_1 -2x_2 + 3x_3\\ 5x_1 + 4x_2 + 2x_3 \\ \end{array}\right) \] \[ T \left(\begin{array}{c} x_1\\ x_2 \\ x_3 \\ x_4 \end{array}\right) = \left(\begin{array}{ccc} -x_1 +3x_2 + x_3 +9x_4\\ 2x_1 + x_3 + 7x_4 \\ 4x_1 + 2x_2 + x_3 + 2x_4 \end{array}\right) \]
Using this theorem, compute the matrix representations of the three linear transformations T,S, and S * T; discover and comment on the relationship between these three matrices.
\[ \left(\begin{array}{cccc} 1 & -2 & 3\\ 5 & 4 & 2 \\ \end{array}\right)* \left(\begin{array}{cccc} -1 & 3 & 1 & 9\\ 2 & 0 & 1 & 7 \\ 4 & 2 & 1 & 2 \end{array}\right)= \left(\begin{array}{cccc} 7 & 9 & 2 & 1\\ 11 & 19 & 11 & 77 \\ \end{array}\right) \]
Commenting : The theorem basically states that if your linear transformation changes the dimensionality, from n to m; then if you apply this transformation to a respective vector, you’ll get a matrix of m x n. I’m not immediately seeing any other relationship, maybe someone can comment and help me out here…Perhaps you can’t apply transformations outside their respective dimensionality; and if you take the product of two transformed vectors; you’ll maintain the lowest dimensionality, are things to extrapolate from the exercise.