Given a linear transformation $T:V \to \mathbb R^{2 \times 2}$,where $V$ is the set of complex numbers over $\mathbb R$ and $\mathbb R^{2 \times 2}$ is the set of $2 \times 2$ matrices with entries in $\mathbb R$ such that for $z=x+iy \in \mathbb C$
$$T(z)=
\begin{bmatrix}
x+7y & 5y\\
-10y & x-7y\\
\end{bmatrix}
\quad$$
- Show that $T$ is an injective linear transformation.
- Describe the range of $T$.
For $\lambda \in \mathbb R$ and $z_1=a+ib$ and $z_2=c+id$ we have that:
$$T(\lambda z_1+z_2)=T((\lambda a+c)+i(\lambda b+d))$$
$$=\begin{bmatrix}
(\lambda a+c)+7(\lambda b+d)& 5(\lambda b+d)\\
-10(\lambda b+d) & (\lambda a+c)-7(\lambda b+d)\\
\end{bmatrix}
\quad$$
$$=\begin{bmatrix}
\lambda a+7\lambda b& 5\lambda b\\
-10\lambda b & \lambda a-7\lambda b\\
\end{bmatrix}
\quad +\begin{bmatrix}
c+7d& 5d\\
-10d & c-7d\\
\end{bmatrix}
\quad$$
$$=\lambda T(z_1)+ T(z_2)$$
Which shows that $T$ is a linear transformation,moreover let $T(z_1)=T(z_2)$,implies that:
$$\begin{bmatrix}
a+7 b& 5 b\\
-10 b & a-7 b\\
\end{bmatrix}
\quad =\begin{bmatrix}
c+7d& 5d\\
-10d & c-7d\\
\end{bmatrix}
\quad$$
$$\implies b=d \implies a=c \implies z_1=z_2$$
And so $T$ is a injection.
Now we know that $\text {ker}(T)=\left\{ 0_V \right\}$ which is helpful since bu rank-nullity theorem:
$$\text{dim}V=\text { dim ker}(T)+\text { dim Im}(T)$$
Which implies that $\text { dim Im}(T)=2$ and hence $\text { Im} (T) \cong V$
What else can be said about the range of $T$? I don't understand what does the author mean by describe
Best Answer
Note that$$T(1)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\quad\text{and that}\quad T(i)=\begin{bmatrix}7&5\\-10&-7\end{bmatrix}.$$So, since $V=\operatorname{span}(\{1,i\})$,\begin{align}\operatorname{range}(T)&=\operatorname{span}\left(\left\{\begin{bmatrix}1&0\\0&1\end{bmatrix},\begin{bmatrix}7&5\\-10&-7\end{bmatrix}\right\}\right)\\&=\left\{\begin{bmatrix}x+7y&5y\\-10y&x-7y\end{bmatrix}\,\middle|\,x,y\in\Bbb R\right\}.\end{align}Note that this is simply to say that $\operatorname{range}(T)=\{T(z)\mid z\in\Bbb C\}$, which is tautological. But the range of $T$ can also be described as$$\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in\Bbb R^{2\times2}\,\middle|\,b=\frac5{14}(a-d)\wedge c=-\frac57(a-d)\right\}.$$