Undergrad Vector visualization of multicollinearity

[Trollfaz]

General linear model is
$$y=a_0+\sum_{i=1}^{i=k} a_i x_i$$
In regression analysis one always collects n observations of y at different inputs of $x_i$s. n>>k or there will be many problems. For each regressor, and response y ,we tabulate all observations in a vector $\textbf{x}_i$ and $\textbf{y}_i$, both is a vector of $R^n$.So multicollinearity is the problem that there's significant correlation between the $x_i$s. In practice some degree of multicollinearity exists. So perfectly no multicollinearity means all the $\textbf{x}_i$ are orthogonal to each other?ie.
$$\textbf{x}_i•\textbf{x}_j=0$$
For different i,j and strong multicollinearity means one of more of the vector makes a very small angle with the subspace form by the other vectors? As far as I know perfect multicollinearity means rank(X)<k. X is a n by k matrix with ith col as $\textbf{x}_i$

 

[scottdave]

Perfect multicollinarity means that at least 1 predictor variable (columns) is a perfect linear combination of one or more of the other variables. Typically the variables are the columns of the matrix and observations are rows. In this situation, the matrix will not be full rank.