Archetype U Archetype U
⬜ Summary Domain is matrices, codomain is column vectors. Domain has dimension 6, while codomain has dimension 4. Cannot be injective, is surjective.
⬜ Definition A linear transformation (Definition LT).
\begin{equation*}
\ltdefn{T}{M_{23}}{\complex{4}},\quad
\lteval{T}{\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}}=
\colvector{a+2b+12c-3d+e+6f\\2a-b-c+d-11f\\a+b+7c+2d+e-3f\\a+2b+12c+5e-5f}
\end{equation*}
⬜ Kernel A basis for the kernel of the linear transformation (Definition KLT).
\begin{equation*}
\set{
\begin{bmatrix}
3 & -4 & 0\\1 & 2 & 1
\end{bmatrix}
,\,
\begin{bmatrix}
-2& -5& 1\\0 & 0 & 0
\end{bmatrix}
}
\end{equation*}
⬜ Injective? Is the linear transformation injective (Definition ILT)? No.
Since the kernel is nontrivial Theorem KILT tells us that the linear transformation is not injective. Also, since the rank cannot exceed 4, we are guaranteed to have a nullity of at least 2, just from checking dimensions of the domain and the codomain. In particular, verify that
\begin{align*}
\lteval{T}{\begin{bmatrix}1&10&-2\\3&-1&1\end{bmatrix}}&=\colvector{-7\\-14\\-1\\-13} & \lteval{T}{\begin{bmatrix}5&-3&-1\\5&3&3\end{bmatrix}}&=\colvector{-7\\-14\\-1\\-13}\text{.}
\end{align*}
This demonstration that \(T\) is not injective is constructed with the observation that
\begin{align*}
\begin{bmatrix}5&-3&-1\\5&3&3\end{bmatrix} &=\begin{bmatrix}1&10&-2\\3&-1&1\end{bmatrix}+\begin{bmatrix}4&-13&1\\2&4&2\end{bmatrix}\\
\end{align*}
and
\begin{align*} \vect{z}&=\begin{bmatrix}4&-13&1\\2&4&2\end{bmatrix}\in\krn{T} \end{align*}so the vector \(\vect{z}\) effectively “does nothing” in the evaluation of \(T\text{.}\)
⬜ 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{\colvector{1\\2\\1\\1},\,
\colvector{2\\-1\\1\\2},\,
\colvector{12\\-1\\7\\12},\,
\colvector{-3\\1\\2\\0},\,
\colvector{1\\0\\1\\5},\,
\colvector{6\\-11\\-3\\-5}}
\end{equation*}
⬜ 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 “nice” 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{
\colvector{1\\0\\0\\0},\,
\colvector{0\\1\\0\\0},\,
\colvector{0\\0\\1\\0},\,
\colvector{0\\0\\0\\1}
}
\end{equation*}
⬜ Surjective? Is the linear transformation surjective (Definition SLT)? Yes.
A basis for the range is the standard basis of \(\complex{4}\text{,}\) so \(\rng{T}=\complex{4}\) and Theorem RSLT tells us \(T\) is surjective. Or, the dimension of the range is 4, and the codomain (\(\complex{4}\)) 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}&=2&\text{domain}&=6
\end{align*}
⬜ Invertible? Is the linear transformation invertible (Definition IVLT, and examine parallels with the existence of matrix inverses.)? No.
The relative sizes of the domain and codomain mean the linear transformation cannot be injective. (Theorem ILTIS)
⬜ Matrix Representation Matrix representation of the linear transformation, as given by Definition MR and explained by Theorem FTMR.
\begin{equation*}
\text{domain basis}=\set{\begin{bmatrix}1&0&0\\0&0&0\end{bmatrix},\,
\begin{bmatrix}0&1&0\\0&0&0\end{bmatrix},\,
\begin{bmatrix}0&0&1\\0&0&0\end{bmatrix},\,
\begin{bmatrix}0&0&0\\1&0&0\end{bmatrix},\,
\begin{bmatrix}0&0&0\\0&1&0\end{bmatrix},\,
\begin{bmatrix}0&0&0\\0&0&1\end{bmatrix}}
\end{equation*}
\begin{equation*}
\text{codomain basis}=\set{\colvector{1\\0\\0\\0},\,
\colvector{0\\1\\0\\0},\,
\colvector{0\\0\\1\\0},\,
\colvector{0\\0\\0\\1}}
\end{equation*}
\begin{equation*}
\text{matrix representation}=\begin{bmatrix}
1 & 2 & 12 & -3 & 1 & 6 \\
2 & -1 & -1 & 1 & 0 & -11 \\
1 & 1 & 7 & 2 & 1 & -3 \\
1 & 2 & 12 & 0 & 5 & -5
\end{bmatrix}
\end{equation*}