MLR doubt regarding invariance of predicted values in the model  $ Y= X\beta +\epsilon  $
when they say the predicted value of y is invariant to full rank linear tranformation on xi's . what does it mean . ?
does it mean that any linear transformation on X  that doesn't change its column rank will give same predicted values ?
 A: For instance, it does not matter for fitted values and residuals if we change the units of measurement of $X$.
Consider transforming $X$ by some invertible $k\times k$ matrix $A$, $XA$ (e.g., change months of schooling to years and meters to centimeters when explaining wages).
This is seen as follows,
\begin{eqnarray*}
P_{XA}&:=&XA\bigl((XA)'XA\bigr)^{-1}(XA)'\\
&=&XA\bigl(A'X'XA\bigr)^{-1}A'X'\\
&=&XAA^{-1}(X'X)^{-1}(A')^{-1}A'X'\\
&=&P_{X}
\end{eqnarray*}
A: Yes.
If you would transform $X$ to $X'$ like: $X'=A^{T}X$, where $A$ is not of full rank, then you would effectively be dropping information in $X$.
Depending on your data and fitting method, this may well result in different predictions for $Y$ after fitting.
As an illustration, consider the situation where $A$ has not full rank, which means that at least 1 of its rows can be written as a linear combination of the others. 
So, we could write for example: $A=BA'$, where
$B=\left(\begin{array}{cccc} 1 & 0 & \cdots & 0\\ 0 & 1 & \, & 0\\ \vdots & \, & \, & \, & \, \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\end{array}\right)$
note the last row, which is $0$.
Substituting this in $X'=A^{T}X$ leads to $X'=\left(BA'\right)^{T}X=A'^{T}B^{T}X$, which will drop the last feature in the data $X$.
