If the Images of Vectors are Linearly Independent, then They Are Linearly Independent

Problem 62

Let $T: \R^n \to \R^m$ be a linear transformation.
Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.

Vectors $\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k$ are linearly independent
if and only if the only solution to the vector equation
\[c_1\mathbf{x}_1+c_2\mathbf{x}_2+\cdots+c_k\mathbf{x}_k=\mathbf{0}_n\]
is $c_1=c_2=\cdots=c_k=0$.

A linear transformation $T:\R^n \to \R^m$ is a map such that

$T(\mathbf{v}+\mathbf{w})=T(\mathbf{v})+\mathbf{w}$ for all $\mathbf{v}, \mathbf{w} \in \R^n$.

$T(c\mathbf{v})=cT(\mathbf{v})$ for all $c \in \R$ and $\mathbf{v}\in \R^n$.

Proof.

Consider a linear combination of vectors in $S$
\[c_1\mathbf{x}_1+c_2\mathbf{x}_2+\cdots+c_k\mathbf{x}_k=\mathbf{0}_n,\]
where $\mathbf{0}_n$ is the $n$ dimensional zero vector.
To show that $S$ is linearly independent, we need to show that the coefficients $c_i$ are all zero.

Thus we have
\begin{align*}
\mathbf{0}_m&=T(\mathbf{0}_n)=T(c_1\mathbf{x}_1+c_2\mathbf{x}_2+\cdots+c_k\mathbf{x}_k)\\
&= c_1T(\mathbf{x}_1)+c_2T(\mathbf{x}_2)+\cdots+c_k T(\mathbf{x}_k).
\end{align*}
In the last step, we used the linearity of $T$.

Since the vectors $T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k)$ are linearly independent, the coefficient of this linear combination of these vectors must be zero.
Thus we have $c_1=c_2=\dots=c_k=0$, hence the set $S$ is linearly independent.

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