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I need help with this question:

Let $T$ be a linear map from $\mathbb{R}^n \to \mathbb{R}^m$, and let $L$ be a line in $\mathbb{R}^n$. Show that $T(L)$ is also a line (is it not enough to just consider lines in the plane).

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A line is a subset of the form $L=\{av+w:a\in \mathbb R\}$ for two fixed vectors $v$ and $w$. By linearity, $T(L)=\{aT(v)+T(w):a\in \mathbb R\}$ is again a space of the same form. Of course it can coincide with just the point $T(w)$ if $T(v)=0$.

The assertion is false. If $T$ is the map defined by $T(v)=0$, then the image of *every* line is a point.

A line is the set of points of the form $x+\alpha v$, for two (fixed) vectors $x,v\in\mathbb{R}$ and $\alpha$ taking all values in $\mathbb{R}$. So the image of $L$ is the set

$$

\{T(x)+\alpha T(v):\alpha\in\mathbb{R}\}

$$

which is a line if and only if $T(v)\ne0$.

A line is of the form $f(t) = \vec{x}_0 + t \vec{d}$ for $t \in \mathbb{R}$.

Thus, $T(f(t)) = T(\vec{x}_0 + t \vec{d}) = T(\vec{x}_0) + t T(\vec{d})$ by linearity, which also has the form of a line.

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