I am reading both LDA lecturer notes and follow the example, both having different covariance calculation.
In LDA lecturer note 1 slide 19,
Did not multiple by number of samples when calculating Covaraince matrix S1, S2.
In LDA lecture note 2' slide 20
She calculate covariance using by multiple the number of samples.
The confusion is most likely due to the n and (n - 1) difference in variance estimators.
$$V_{biased} = \dfrac{\sum_{i=1}^{n} (z_i - \mu_z)^2}{n}$$ $$V_{unbiased} = \dfrac{\sum_{i=1}^{n} (z_i - \mu_z)^2}{n - 1}$$
On slide 9 you define scatter as biased variance estimator without dividing by n: $$S = \sum_{i=1}^{n} (z_i - \mu_z)^2 = n ((\sum_{i=1}^{n} (z_i - \mu_z)^2) / n) = n V_{biased}[z]$$
You can write an equivalent formula based on unbiased variance estimator: $$S = \sum_{i=1}^{n} (z_i - \mu_z)^2 = (n - 1) ((\sum_{i=1}^{n} (z_i - \mu_z)^2) / (n - 1)) = (n - 1) V_{unbiased}[z]$$
On slide 20 you use the scatter $S_1$ based on the covariance matrix $cov(c_1)$ that is calculated by some software that uses the $V_{unbiased}$ formula:
$$S_1 = (n_1 - 1) V_{unbiased}[z_1] = 5 * \left[ {\begin{array}{cc} 10 & 8 \\ 8 & 7.2 \\ \end{array} } \right] $$
You can check that it all adds up in any computer language or manually, e.g. here is R:
> c1 = t(matrix(c(1, 2, 2, 3, 3, 3, 4, 5, 5, 5), nrow = 2))
> (nrow(c1) - 1) * var(c1)
[,1] [,2]
[1,] 10 8.0
[2,] 8 7.2
> # First element of the scatter matrix calculated manually
> sum((c1[, 1] - mean(c1[, 1])) ^ 2)
[1] 10
