Task:
$X_1=(-2,6,3,-1)^T$
$X_2=(7,-3,-6,2)^T$
$X_3=(3,9,0,0)^T$
$Y=(2,-1,3,8)^T$
I need to find projection to Z, which is a linear projection. $Z$ should be expressed as $X_1,X_2$ and $X_3 $ combination. So basically I need to find distance between $Y$ and $L$. So I need to find $\hat{\beta}$
My solution:
First Let'S say that
$X=( X_1,X_2,X_3) $
I need to find $(X^TX)^{-1}$,
$$X^TX = \begin{bmatrix} 50 & -52 & 48\\ -52 & 98 & -6 \\ 48 & -6 & 90 \end{bmatrix}$$
But I cannot find $(X^TX)^{-1}$ because it is not invertible.
Don't know where Can I get from here.