Let $T : \mathbb V \to \mathbb W$ be a linear transformation from a vector space $\mathbb V$ into a vector space $\mathbb W$. Prove that the range of $T$ is a subspace of $\mathbb W$.
OK here is my attempt…
If we let $x$ and $y$ be vectors in $\mathbb V$, then the transformation of these vectors will look like this… $T(x)$ and $T(y)$.
If we let $\mathbb V$ be a vector space in $\mathbb R^3$ and $\mathbb W$ be a vector space in $\mathbb R^2$, then
$$
T \begin{pmatrix} x_1\\ x_2 \\ x_3
\end{pmatrix} = T\begin{pmatrix} x_1 + 2x_2 \\ 3x_3 + 4 \end{pmatrix}.
$$
Now if we tried to row reduce the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$
we would get $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} .$
SO the range of $T$ are the linear combinations of the pivot colums of the matrix above.
This is as much as I can do by myself. But now that i think about it, I believe this is wrong because the linear combinations of the pivot columns will give out any vector in $\mathbb R^2$, and not in the subspace of $\mathbb W$.
Any help will be appreciated.
Best Answer
Given any two vector spaces $\mathbb{V, W}$ over $F$, and $T:\mathbb{V}\to\mathbb{W}$ is a linear transformation.
The 1. and 2. complete the proof.
$\dagger:$ linearity of $T:\mathbb{V}\to\mathbb{W}$.