MHB Proving Linear Maps are One-to-One

[baseball3030]

One to One

Prove that A linear map T: Rn->Rm is one to one:

Again, the only thing I can think of doing is possibly using rank nullity theorem but then again I think this can be proved by using independence assumption.

 

[Nono713]

Hint: find a basis of $\mathbb{R}^n$ (the columns of $I_n$ will do) and show that the image of this basis under $T$ cannot be linearly independent in $\mathbb{R}^m$. Use this to show that $\mathbf{0} \in \mathrm{R}^m$ has two preimages under $T$.​

 

[Deveno]

From a previous topic, if $B$ is any basis for $\Bbb R^n$, then $T(B)$ is linearly independent if $T$ is 1-1.

Thus $\text{dim}(\text{im}(T)) = |T(B)| = n$.

Since any basis $C$ of $\Bbb R^m$ is a MAXIMAL linearly independent set, we have $n \leq m$, contradicting $m < n$.

Our only assumption was that $T$ was 1-1, so this cannot be the case.

(Bacterius' approach is good, too