K fold cross validation clarification I don't know if I understood K-fold CV correctly. Here I will explain briefly what I am trying to do. 
I have a matrix X which I split into 3 folds, 2 training $X_1$, $X_2$ and one testing $X_3$: 

Step 1


*

*I concatenate $X_1$ and $X_2$ so I have $X_{1,2}$, then I generate a $Y_{1,2}$. I compute the $\hat{\beta_{1,2}}$ from the formula: $\hat{\beta_{1,2}}=(X^tX+\lambda I_p)^{-1}X^t Y$ and I compute the predicted $\hat{Y_{1,2}}=X_{1,2}\hat{\beta_{1,2}}$ 


step 2 I reshuffle. Then I have:

I do the same as in step 1, I compute $X_{1,3}$, $Y_{1,3}$,  $\hat{\beta_{1,3}}$ and $\hat{Y_{1,3}}$
Then I reshuffle again and get:

And I compute again:  $X_{2,3}$, $Y_{2,3}$,  $\hat{\beta_{2,3}}$ and $\hat{Y_{2,3}}$
step 3 Computing the error (This is the part where I'm not quite sure if it is correct)
The formula for the square loss is: $L(Y)=(Y-\hat{Y})^2$ 
Would then these be correct:
$L(Y_1)=(Y_{1,2}-\hat{Y_{2,3}})^2$, 
$L(Y_2)=(Y_{1,3}-\hat{Y_{2,3}})^2$,
$L(Y_3)=(Y_{2,3}-\hat{Y_{1,2}})^2$  
step 4 Calculating the mean of the above errors
 $CV = \frac{1}{n}\sum_j^n[L(Y)]$
 A: I find your notation a bit hard but I think I get it, additionally I think you are slightly off. 
Once you have calculated $\hat \beta_{12}$ you want to use this to create predicted values for the values of $Y$ that were not in your training data, Lets call this $\hat Y_{3 \mid 1,2}$ . For clarity that is to say the predicted values of $Y_3$ after fitting a model using data from $X_{12}$ . You then calculate your error as $ (Y_3 - \hat Y_{3 \mid 1,2}) ^ 2$. Your last line is then correct that you then repeat this for all your folds of data and take the mean across all your predicted errors. 
A: Let's do a mini example with 6 samples:
Let $X_1$, $X_2$ and $X_3$ be three sets:
$$
X_1 = \{x_1,x_2\}\\
X_2 = \{x_3,x_4\}\\
X_3 = \{x_5,x_6\}\\
$$
Each $x_i$ here is a vector. Judging from your previous question, this will be a 1x15 vector.
You also have:
$$
Y_1 = \{y_1, y_2\}\\
Y_2 = \{y_3, y_4\}\\
Y_3 = \{y_5, y_6\}\\
$$
Those $y_i$ are single values.
Then you create three sets by merging those three folds in different combinations:
$$
X_{1,2} = X_1 \cup X_2\\
X_{1,3} = X_1 \cup X_3\\
X_{2,3} = X_2 \cup X_3
$$
Once you train the three models you have $\beta_{1,2}$, $\beta_{1,3}$ and $\beta_{2,3}$.
Here is the important part.
To calculate the predictions you do:
$$
\hat{Y_1} = X_1\beta_{2,3}
$$
Similarly for the other two:
$$
\hat{Y_2} = X_2\beta_{1,3}\\
\hat{Y_3} = X_3\beta_{1,2}
$$
Then your error is:
$$
L(y_1) = (y_1 - \hat{y_1})^2\\
L(y_2) = (y_2 - \hat{y_2})^2\\
L(y_3) = (y_3 - \hat{y_3})^2\\
L(y_4) = (y_4 - \hat{y_4})^2\\
L(y_5) = (y_5 - \hat{y_5})^2\\
L(y_6) = (y_6 - \hat{y_6})^2\\
CV=\frac{1}{n}\sum_i[L(y_i)]
$$
Note that $\hat{y_1}$ and $\hat{y_2}$ are samples from your matrix $\hat{Y_1}$. $\hat{y_3}$ and $\hat{y_4}$ are part of $\hat{Y_2}$. $\hat{y_5}$ and $\hat{y_6}$ is in $\hat{Y_3}$.
