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Archetype V Archetype V

⬜  Summary   Domain is polynomials, codomain is matrices. Both domain and codomain have dimension 4. Injective, surjective, invertible. Square matrix representation, but domain and codomain are unequal, so no eigenvalue information.
⬜  Definition  A linear transformation (Definition LT).
\begin{equation*} \ltdefn{T}{P_3}{M_{22}},\quad\lteval{T}{a+bx+cx^2+dx^3}= \begin{bmatrix} a+b & a-2c\\ d & b-d \end{bmatrix} \end{equation*}
⬜  Kernel  A basis for the kernel of the linear transformation (Definition KLT).
\begin{equation*} \set{\ } \end{equation*}
⬜  Injective?  Is the linear transformation injective (Definition ILT)? Yes.

Since the kernel is trivial Theorem KILT tells us that the linear transformation is injective.

⬜  Spanning Set for Range  A spanning set for the range of a linear transformation (Definition RLT) can be constructed easily by evaluating the linear transformation on a standard basis (Theorem SSRLT).
\begin{equation*} \set{\begin{bmatrix}1&1\\0&0\end{bmatrix},\, \begin{bmatrix}1&0\\0&1\end{bmatrix},\, \begin{bmatrix}0&-2\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\1&-1\end{bmatrix}} \end{equation*}
‚¨ú¬†¬†Range¬†¬†A basis for the range of the linear transformation (Definition¬†RLT). If the linear transformation is injective, then the spanning set just constructed is guaranteed to be linearly independent (Theorem¬†ILTLI) and is therefore a basis of the range with no changes. Injective or not, this spanning set can be converted to a ‚Äúnice‚ÄĚ linearly independent spanning set by making the vectors the rows of a matrix (perhaps after using a vector representation), row-reducing, and retaining the nonzero rows (Theorem¬†BRS), and perhaps un-coordinatizing.
\begin{equation*} \set{ \begin{bmatrix}1&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&1\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\1&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&1\end{bmatrix} } \end{equation*}
⬜  Surjective?  Is the linear transformation surjective (Definition SLT)? Yes.

A basis for the range is the standard basis of \(M_{22}\text{,}\) so \(\rng{T}=M_{22}\) and Theorem RSLT tells us \(T\) is surjective. Or, the dimension of the range is 4, and the codomain (\(M_{22}\)) has dimension 4. So the transformation is surjective.

⬜  Subspace Dimensions  Subspace dimensions associated with the linear transformation (Definition ROLT, Definition NOLT). Verify Theorem RPNDD, and examine parallels with earlier results for matrices.
\begin{align*} \text{rank}&=4&\text{nullity}&=0&\text{domain}&=4 \end{align*}
⬜  Invertible?  Is the linear transformation invertible (Definition IVLT, and examine parallels with the existence of matrix inverses.)? Yes.

Both injective and surjective (Theorem ILTIS). Notice that since the domain and codomain have the same dimension, either the transformation is both injective and surjective (making it invertible, as in this case) or else it is both not injective and not surjective.

⬜  Matrix Representation  Matrix representation of the linear transformation, as given by Definition MR and explained by Theorem FTMR.
\begin{equation*} \text{domain basis}=\set{1,\,x,\,x^2,\,x^3} \end{equation*}
\begin{equation*} \text{codomain basis}=\set{\begin{bmatrix}1&0\\0&0\end{bmatrix},\, \begin{bmatrix}0&1\\0&0\end{bmatrix},\, \begin{bmatrix}0&0\\1&0\end{bmatrix},\, \begin{bmatrix}0&0\\0&1\end{bmatrix}} \end{equation*}
\begin{equation*} \text{matrix representation}=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 0 & -2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \end{bmatrix} \end{equation*}
⬜  Inverse Linear Transformation  The inverse linear transformation (Definition IVLT). Verify that \(T^{-1}\left(T\left(\vect{x}\right)\right)=\vect{x}\) and \(T\left(T^{-1}\left(\vect{x}\right)\right)=\vect{x}\text{.}\)
\begin{equation*} \ltdefn{\ltinverse{T}}{M_{22}}{P_3},\quad\lteval{\ltinverse{T}}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}=(a - c - d)+ (c + d)x +\frac{1}{2}(a - b - c - d)x^2+cx^3 \end{equation*}