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### Solution: Linear Discriminant Projection To compute the Linear Discriminant Analysis (LDA) projection, we'll follow these steps: --- ### Step 1: Define the Two Classes - Class \(X_1\): \[ X_1 = \{(4, 2), (2, 2), (3, 2), (3, 5), (3, 4)\} \] - Class \(X_2\): \[ X_2 = \{(8, 7), (9, 6), (7, 7), (9, 8), (10, 9)\} \] --- ### Step 2: Compute the Mean of Each Class For Class \(X_1\): \[ \mu_1 = \left( \frac{4 + 2 + 3 + 3 + 3}{5}, \frac{2 + 2 + 2 + 5 + 4}{5} \right) \] \[ \mu_1 = \left( \frac{15}{5}, \frac{15}{5} \right) = (3, 3) \] For Class \(X_2\): \[ \mu_2 = \left( \frac{8 + 9 + 7 + 9 + 10}{5}, \frac{7 + 6 + 7 + 8 + 9}{5} \right) \] \[ \mu_2 = \left( \frac{43}{5}, \frac{37}{5} \right) = (8.6, 7.4) \] --- ### Step 3: Compute the Scatter Matrices #### Within-Class Scatter Matrix \(S_W\) For Class ______ __________ _________ ________ _________ ______ ________.
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