Linear Transformation. Adjoint operator.

Question

I found a problem while I was working out an exercise:

If the linear transformation $A: \mathbb{R}^{m} \longrightarrow \mathbb{R}^{n}$ that transform an orthonormal set of $\mathbb{R}^{m}$ in an orthonormal set of $\mathbb{R}^{n}$, the adjoint $A^{\ast}: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}$ is left inverse of $A$?

I'm not sure about the injectivity. Any hint? I didn't want a solution.

Answer 1

For injectivity, note that for any non-zero vector $x$, the normalization $\hat x := \dfrac x {\|x\|}$ is well-defined. Then, the set $\{\hat x\}$ is orthonormal. What then can we say about $\{A\hat x\}$? And then $Ax$?

So, what can we say about $\ker A$?

Answer 2

Pick $\{e_i\}$ an orthonormal basis for $\mathbb{R}^m$. The hypothesis gives you that $$\langle Ae_i, Ae_j\rangle =\langle e_i,e_j \rangle. $$ Use this to establish that $$\langle Av,Aw \rangle= \langle v,w \rangle $$ for any $v,w \in \mathbb{R}^m$. It then holds that $$\langle A^*Av,w \rangle=\langle v, w \rangle$$ for all $v,w \in \mathbb{R}^m$, which should give you everything you need to conclude what you want.