Each principal component is a straight-line (linear) combination of the original features?

Each principal component is a straight-line (linear) combination of the original features?

• A. Nonlinear combination
• B. Random combination
• C. Straight-line (linear) combination
• D. Piecewise combination

Answer: (C)

The main idea is that PCA creates new features as linear mixtures of the original ones. After centering the data, PCA identifies directions in feature space that capture the most variance; these directions are given by the eigenvectors of the covariance matrix. A principal component is the projection of the data onto one of these directions, which mathematically is a dot product z = w^T x. That means each principal component is a weighted sum of the original features with fixed weights, i.e., a linear (straight-line) combination. This is not nonlinear or piecewise; nonlinear variants require different methods like kernel PCA. So the statement is true: each principal component is a linear combination of the original features.

The main idea is that PCA creates new features as linear mixtures of the original ones. After centering the data, PCA identifies directions in feature space that capture the most variance; these directions are given by the eigenvectors of the covariance matrix. A principal component is the projection of the data onto one of these directions, which mathematically is a dot product z = w^T x. That means each principal component is a weighted sum of the original features with fixed weights, i.e., a linear (straight-line) combination. This is not nonlinear or piecewise; nonlinear variants require different methods like kernel PCA. So the statement is true: each principal component is a linear combination of the original features.