The problem is: Let $A$ be a $m\times n$ matrix. Show that if $A^*A=I$, the $n\times n$ identity matrix ($A^*$ is the adjoint), then the columns $g_1,\cdots,g_n$ in $A$ constitute an orthonormal system in $\mathbb{C}^m$.
What the adjoint of $A$ is? And how could this be true in $\mathbb{C}^m$? And by the way, I want to know what's the relationship, or connection between the columns and rows of $A$? Cause this is a knowledge for the frame analysis, I especially want to know what is $<A_k,A_k>$? (The dot product between the same row of $A$.) Thanks in advance.
Assume $A^*A=I$. On both sides of this equation is an $n\times n$ matrix. We can understand what this equation means, by writing down what it means for the $(i,j)$th component on both sides:
$$\sum_{k=1}^m \overline{A}_{ki}A_{kj}=\delta^i_j$$
where $\delta^i_j$ is the Kronecker delta, which is $1$ if $i=j$ and $0$ otherwise, and $A_{ij}$ is the $(i,j)$th component of the matrix $A$, i.e. the number at row $i$, column $j$.
The columns of $A$ are the vectors $g_j=(A_{ij})_{i=1,..,m}\in\mathbb{C}^m$ for $j=1,..,n$. The above equation can be rewritten using the scalar product in $\mathbb{C}^m$:
$$\langle g_i, g_j\rangle = \sum_{k=1}^m \overline{A}_{ki}A_{kj} = \delta^i_j$$
where $\langle v,w\rangle = \sum_{k=1}^m \overline{v}_k w_k$ for $v,w\in\mathbb{C}^m$. $\langle\cdot,\cdot\rangle$ is called the scalar product. This means by definition that $(g_n)_n$ is an orthonormal system of $\mathbb{C}^m$: $$\|g_i\|^2=\langle g_i, g_i\rangle=1\,\,\text{and}\,\,\langle g_i,g_j\rangle=0\,\,\text{for}\,\,i\not=j$$
That is, all the column vectors have norm $1$ and are orthogonal to one another.