PCA selects low-dimensional linear surfaces to maximize the captured variance.

PCA selects low-dimensional linear surfaces to maximize the captured variance.

• A. PCA reduces dimensionality by projecting data onto a nonlinear surface.
• B. PCA reduces dimensionality by projecting data onto a linear surface that maximizes explained variance.
• C. PCA cannot be used for visualization.
• D. PCA guarantees perfect reconstruction.

Answer: (B)

PCA is about finding a linear subspace of lower dimension that captures as much variation in the data as possible. It does this by projecting data onto the span of the principal components, which are orthogonal directions chosen to maximize the variance of the projected data (explained variance). The representation you get is a projection onto a linear surface, and the components are ordered so the first captures the most variance, the next captures the most of what remains, and so on. This is why you can use PCA to visualize high‑dimensional data by plotting the first two or three components. Perfect reconstruction is not guaranteed in general—only if you keep all components or if the data lie exactly on the chosen linear subspace.

PCA is about finding a linear subspace of lower dimension that captures as much variation in the data as possible. It does this by projecting data onto the span of the principal components, which are orthogonal directions chosen to maximize the variance of the projected data (explained variance). The representation you get is a projection onto a linear surface, and the components are ordered so the first captures the most variance, the next captures the most of what remains, and so on. This is why you can use PCA to visualize high‑dimensional data by plotting the first two or three components. Perfect reconstruction is not guaranteed in general—only if you keep all components or if the data lie exactly on the chosen linear subspace.